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ALTERNATING AND SYMMETRIC SEPARABILITY OF FREE PRODUCTS

Part of: Special aspects of infinite or finite groups Low-dimensional topology Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  16 June 2026

DONGXIAO ZHAO Open the ORCID record for DONGXIAO ZHAO [Opens in a new window]  and
QIANG ZHANG Open the ORCID record for QIANG ZHANG [Opens in a new window]
DONGXIAO ZHAO
Affiliation:
School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, PR China e-mail: zdxmath@stu.xjtu.edu.cn
QIANG ZHANG*
Affiliation:
School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, PR China
*
e-mail: zhangq.math@mail.xjtu.edu.cn
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Abstract

Let $F \ast G$ be a free product of a free group F and a LERF group G. We provide sufficient conditions for a subgroup H of $F \ast G$ to be $\mathcal A\cup \mathcal S$-separable, that is, for any finite set $\{\gamma _1, \ldots , \gamma _n\} \subset (F \ast G) \setminus H$, there is a surjection f from $F \ast G$ to an alternating or symmetric group such that $f(\gamma _i) \notin f(H)$ for all i. As a corollary, any finitely generated infinite-index subgroup of a free group is $\mathcal A\cup \mathcal S$-separable in the free product of the free group and an arbitrary LERF group, generalising a result of Wilton [‘Alternating quotients of free groups’, Enseign. Math. 58 (2012), 49–60].

Keywords

locally extended residually finitealternating and symmetric separablefree productprecoverrelative Cayley graph

MSC classification

Primary: 20F65: Geometric group theory
Secondary: 20E26: Residual properties and generalizations; residually finite groups 57M07: Topological methods in group theory

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 114 , Issue 3 , December 2026 , pp. 501 - 515
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Copyright
© The Author(s), 2026. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

1 Introduction

Residual properties have been widely discussed for infinite discrete groups. Let $\mathcal C$ script upper C be a class of groups. A group G is said to be residually $\mathcal C$ script upper C if for every $g \in G\setminus \{e\}$ g element of upper G minus StartSet e EndSet , there exists a homomorphism $f: G \to C$ f colon upper G right arrow upper C onto some group $C \in \mathcal C$ upper C element of script upper C such that $f(g)$ f left parenthesis g right parenthesis is nontrivial. The property of residual finiteness admits several generalisations, most notably the concept of LERF: a group G is said to be locally extended residually finite (LERF) if for any finitely generated subgroup $H \leq G$ upper H less than or equals upper G and any $\gamma \in G \setminus H$ gamma element of upper G minus upper H , there is a homomorphism f from G onto a finite group such that $f(\gamma )\notin f(H)$ f left parenthesis gamma right parenthesis not an element of f left parenthesis upper H right parenthesis .

The LERF-ness of groups is a property closely related to low-dimensional topology and has been widely studied. Large classes of groups are known to be LERF: for example, finitely generated free groups [Reference Hall6], closed orientable hyperbolic surface groups [Reference Scott13], compact Seifert fibre space groups [Reference Scott13], Sol manifold groups [Reference Mal’cev8] and hyperbolic 3-manifold groups [Reference Agol1, Reference Wise19]. In fact, by [Reference Sun15], a compact (with empty or tori boundary) orientable irreducible 3-manifold M has $\pi _1(M)$ pi 1 left parenthesis upper M right parenthesis LERF if and only if M supports one of Thurston’s eight geometries. Moreover, by [Reference Wilton17], limit groups are LERF, which generalises both the cases of free groups and surface groups. However, nontrivial graph manifold groups are not LERF [Reference Niblo and Wise11], noncompact arithmetic hyperbolic m-manifold groups with $m>3$ m greater than 3 , and compact arithmetic hyperbolic m-manifold groups with $m>4$ m greater than 4 (except when defined by octonions with $m=7$ m equals 7 ) are not LERF [Reference Sun15].

The LERF-ness admits a broader generalisation through the restriction of its finite quotients to a more specific class, for example, the class of alternating groups $\mathcal A$ script upper A and symmetric groups $\mathcal S$ script upper S .

Definition 1.1. Let H be a subgroup of a finitely generated group G and let $\mathcal C$ script upper C be a class of groups. We say that H is $\mathcal C$ script upper C -separable in G if for any finite set $\{\gamma _1, \ldots , \gamma _n\} \subset G \setminus H$ StartSet gamma 1 comma ellipsis comma gamma Subscript n Baseline EndSet subset of upper G minus upper H , there is a surjective homomorphism f from G to a group in $\mathcal C$ script upper C such that $f(\gamma _i) \notin f(H)$ f left parenthesis gamma Subscript i Baseline right parenthesis not an element of f left parenthesis upper H right parenthesis for all i.

Note that $\mathcal C$ script upper C -separability is a property of a certain subgroup. In particular, if the trivial subgroup of G is $\mathcal C$ script upper C -separable, then G is residually $\mathcal C$ script upper C .

Notation. Throughout this paper, $F_r$ upper F Subscript r is a free group of rank $r>1$ r greater than 1 and $H^g=gHg^{-1}$ upper H Superscript g Baseline equals g upper H g Superscript negative 1 is the conjugation of a subgroup H by g. For a set V, $|V|$ StartAbsoluteValue upper V EndAbsoluteValue denotes the cardinality of V.

In [Reference Wilton18], Wilton showed that the finitely generated infinite-index subgroups of free groups are $\mathcal A$ script upper A -separable.

Theorem 1.2 [Reference Wilton18, Theorem A]

Let H be a finitely generated subgroup of infinite index in the free group $F_r$ upper F Subscript r and let $\{\gamma _1, \ldots , \gamma _n\}$ StartSet gamma 1 comma ellipsis comma gamma Subscript n Baseline EndSet be any finite set of $F \setminus H$ upper F minus upper H . Then, there is a surjection f from $F_r$ upper F Subscript r to some finite alternating group $A_k$ upper A Subscript k such that $f(\gamma _i) \notin \, f(H)$ f left parenthesis gamma Subscript i Baseline right parenthesis not an element of f left parenthesis upper H right parenthesis for all i.

Wilton also conjectured that the conclusion of Theorem 1.2 holds for closed orientable hyperbolic surface groups. In 2021, Buran [Reference Buran2] confirmed Wilton’s conjecture. Moreover, he generalised this result to right-angled Coxeter groups, and provided a complete characterisation of the $\mathcal A$ script upper A -separability and the $\mathcal S$ script upper S -separability of convex-cocompact subgroups.

We investigate the residual properties and the $\mathcal C$ script upper C -separability of free products of groups. It is well known that both the residual finiteness and LERF-ness are preserved under taking free products (see [Reference Burns3, Reference Romanovskii12]). In 1984, Tamburini and Wilson [Reference Tamburini and Wilson16] studied the residually alternating property of the free products of residually finite groups.

Theorem 1.3 [Reference Tamburini and Wilson16, Theorem 1]

Let G be a free product of at least two nontrivial residually finite groups. Then, G is residually alternating unless G is the infinite dihedral group $\mathbb Z/2\mathbb Z\ast \mathbb Z/2\mathbb Z$ double struck upper Z divided by 2 double struck upper Z asterisk double struck upper Z divided by 2 double struck upper Z .

Note that LERF groups are residually finite and if the trivial subgroup of G is $\mathcal A$ script upper A -separable, then G is residually alternating. It is natural to ask the following question.

Question 1.4. Which kinds of subgroups are $\mathcal A$ script upper A -separable or $\mathcal S$ script upper S -separable in free products of LERF groups?

By combining the methods in [Reference Gitik5, Reference Wilton18], we show that there exists a class of $\mathcal A \cup \mathcal S$ script upper A union script upper S -separable subgroups in the free product of a LERF group with a free group $F_r$ upper F Subscript r of rank $r>1$ r greater than 1 .

Theorem 1.5. Let G be a LERF group and let $H\leq F_r\ast G$ upper H less than or equals upper F Subscript r Baseline asterisk upper G be a finitely generated subgroup. Then, H is $\mathcal A \cup \mathcal S$ script upper A union script upper S -separable in $F_r\ast G$ upper F Subscript r Baseline asterisk upper G if one of the following holds:

  1. (1) $H \cap F_r^{\gamma }=1$ upper H intersection upper F Subscript r Superscript gamma Baseline equals 1 for every $\gamma \in F_r \ast G$ gamma element of upper F Subscript r Baseline asterisk upper G ; or

  2. (2) $H \cap F_r^{\gamma } \neq 1$ upper H intersection upper F Subscript r Superscript gamma Baseline not equals 1 and has infinite index in $F_r^{\gamma }$ upper F Subscript r Superscript gamma for some $\gamma \in F_r \ast G$ gamma element of upper F Subscript r Baseline asterisk upper G .

Since every finitely generated infinite-index subgroup of $F_r$ upper F Subscript r satisfies the hypothesis in Theorem 1.5, the following immediate corollary represents a generalisation of Theorem 1.2 regarding the $\mathcal A$ script upper A -separability of free groups.

Corollary 1.6. Any finitely generated infinite-index subgroup of $F_r$ upper F Subscript r is $\mathcal A \cup \mathcal S$ script upper A union script upper S -separable in the free product $F_r \ast G$ upper F Subscript r Baseline asterisk upper G , where G is a LERF group.

Note that $\mathcal A \cup \mathcal S$ script upper A union script upper S -separable means the quotient is required to be either an alternating or a symmetric group. Since G is an arbitrary LERF group, no uniform modification method seems to exist that would guarantee the quotient is precisely an alternating group. Moreover, the hypothesis of ‘infinite index’ is essential. The following example shows there exist finite-index subgroups that are not alternating separable in such a free product.

Example 1.7. Let $F_2=\langle a,b \mid - \rangle $ upper F 2 equals left angle bracket a comma b vertical bar minus right angle bracket , $G=\langle c \mid c^2\rangle $ upper G equals left angle bracket c vertical bar c squared right angle bracket . We consider $\phi : F_2 \ast G \to \mathbb Z/2\mathbb Z$ phi colon upper F 2 asterisk upper G right arrow double struck upper Z divided by 2 double struck upper Z given by $\phi (a)=\phi (b)=1$ phi left parenthesis a right parenthesis equals phi left parenthesis b right parenthesis equals 1 , $\phi (c)=0$ phi left parenthesis c right parenthesis equals 0 . Then, $\ker \phi $ kernel phi is a finitely generated normal subgroup of $F_2\ast G$ upper F 2 asterisk upper G of index 2. For any surjection $f: F_2\ast G \to A_n$ f colon upper F 2 asterisk upper G right arrow upper A Subscript n Baseline , $f(\ker \phi )$ f left parenthesis kernel phi right parenthesis is normal in $A_n$ upper A Subscript n with index at most 2. If $n=3$ n equals 3 or $n\geq 5$ n greater than or equals 5 , then $A_n$ upper A Subscript n is simple and, hence, $f(\ker \phi )=A_n$ f left parenthesis kernel phi right parenthesis equals upper A Subscript n ; if $n=4$ n equals 4 , then $f(\ker \phi )$ f left parenthesis kernel phi right parenthesis is again $A_n$ upper A Subscript n because the only proper normal subgroup of $A_4$ upper A 4 is of index 3. Thus, $\ker \phi $ kernel phi is not $\mathcal A$ script upper A -separable.

To prove Theorem 1.5, we construct a relative Cayley graph of $F_r \ast G$ upper F Subscript r Baseline asterisk upper G . This graph corresponds to the 1-skeleton of a primary sheeted cover of the representation complex associated with $F_r \ast G$ upper F Subscript r Baseline asterisk upper G . The action of $F_r \ast G$ upper F Subscript r Baseline asterisk upper G on its vertex set is then used to demonstrate $\mathcal A \cup \mathcal S$ script upper A union script upper S -separability.

This paper is organised as follows. In Section 2, we review some topological tools, such as labelled graphs and precovers, to prepare for proving the main results. In Section 3, we review the specific construction of Wilton in [Reference Wilton18] and bring in Jordan’s theorem. In Section 4, we introduce the Markus-Epstein algorithm to provide a more concrete characterisation of the structure of subgroups defined by labelled graphs. Finally, in Section 5, we employ Wilton’s construction and the Markus-Epstein algorithm to prove Theorem 1.5.

2 Preliminaries

In this section, we review some notions and properties of LERF-ness and the topology of graphs. Let $G=\langle X \mid R \rangle $ upper G equals left angle bracket upper X vertical bar upper R right angle bracket be a group and

$$ \begin{align*}X^\ast:=\{x,x^{-1} \mid x \in X\}.\end{align*} $$ upper X Superscript asterisk Baseline colon equals StartSet x comma x Superscript negative 1 Baseline vertical bar x element of upper X EndSet period

2.1 Labelled graphs and embedding

A graph is a one-dimensional CW-complex consisting of two sets $\mathrm {V}$ normal upper V and $\mathrm {E}$ normal upper E representing the 0-cells and 1-cells, respectively, and two functions $\mathrm {E} \to \mathrm {E}$ normal upper E right arrow normal upper E and $\mathrm {E} \to \mathrm {V}$ normal upper E right arrow normal upper V . We call $e \in \mathrm {E}$ e element of normal upper E an edge and $v \in \mathrm {V}$ v element of normal upper V a vertex. For each $e \in \mathrm {E}$ e element of normal upper E , there is an element $\bar {e} \in \mathrm {E}$ e overbar element of normal upper E and an element $\iota (e)\in \mathrm V$ iota left parenthesis e right parenthesis element of normal upper V , satisfying $\bar {\bar {e}}=e$ e overbar overbar equals e and $\bar {e} \neq e$ e overbar not equals e . A map of a graph consists of a pair of functions preserving the structure, and mapping edges to edges and vertices to vertices.

A graph $\Gamma $ normal upper Gamma is called a labelled graph (with the group $G=\langle X \mid R \rangle $ upper G equals left angle bracket upper X vertical bar upper R right angle bracket ) if it is equipped with a labelling function, $\mathrm {Lab}: \mathrm {E}(\Gamma ) \to X^\ast $ upper L a b colon normal upper E left parenthesis normal upper Gamma right parenthesis right arrow upper X Superscript asterisk , satisfying $\mathrm {Lab}(\bar {e})=\mathrm {Lab}(e)^{-1}$ upper L a b left parenthesis e overbar right parenthesis equals upper L a b left parenthesis e right parenthesis Superscript negative 1 and the immersion condition: $e_1=e_2$ e 1 equals e 2 provided $\mathrm {Lab}(e_1)=\mathrm {Lab}(e_2)$ upper L a b left parenthesis e 1 right parenthesis equals upper L a b left parenthesis e 2 right parenthesis and $\iota (e_1)=\iota (e_2)$ iota left parenthesis e 1 right parenthesis equals iota left parenthesis e 2 right parenthesis . The label of a path $p=e_1 e_2\cdots e_n \subset \Gamma $ p equals e 1 e 2 midline horizontal ellipsis e Subscript n Baseline subset of normal upper Gamma is defined by $\mathrm {Lab}(p) \equiv \mathrm {Lab}(e_1)\cdot \mathrm {Lab}(e_2)\cdots \mathrm {Lab}(e_n)\in G$ upper L a b left parenthesis p right parenthesis identical to upper L a b left parenthesis e 1 right parenthesis dot upper L a b left parenthesis e 2 right parenthesis midline horizontal ellipsis upper L a b left parenthesis e Subscript n Baseline right parenthesis element of upper G .

If, in addition, any path p with $\mathrm {Lab}(p)=1$ upper L a b left parenthesis p right parenthesis equals 1 is closed, then $\Gamma $ normal upper Gamma is called G-based. For each vertex v and each word $g \equiv x_1 x_2 \cdots x_n$ g identical to x 1 x 2 midline horizontal ellipsis x Subscript n Baseline , since $\Gamma $ normal upper Gamma is a labelled graph, there is a unique path p in $\Gamma $ normal upper Gamma originating from v with $\mathrm {Lab}(p) \equiv x_1 x_2 \cdots x_n$ upper L a b left parenthesis p right parenthesis identical to x 1 x 2 midline horizontal ellipsis x Subscript n Baseline . The subgroup

$$ \begin{align*}\mathrm{Lab}(\Gamma, v):=\{\mathrm{Lab}(p) \mid p~ \textrm{is a loop in} ~\Gamma \ \textrm{with base point} ~v\}\leq G\end{align*} $$ StartLayout 1st Row upper L a b left parenthesis normal upper Gamma comma v right parenthesis colon equals StartSet upper L a b left parenthesis p right parenthesis vertical bar p is a loop in normal upper Gamma with base point v EndSet less than or equals upper G EndLayout

is called the subgroup of G determined by the graph $\Gamma $ normal upper Gamma .

Let $x \in X^\ast $ x element of upper X Superscript asterisk . A vertex $v \in \mathrm {V}(\Gamma )$ v element of normal upper V left parenthesis normal upper Gamma right parenthesis is called x-saturated if there exists $e\in \mathrm {E}(\Gamma )$ e element of normal upper E left parenthesis normal upper Gamma right parenthesis with $\iota (e)=v$ iota left parenthesis e right parenthesis equals v and $\mathrm {Lab}(e)=x$ upper L a b left parenthesis e right parenthesis equals x . If $\Gamma $ normal upper Gamma is x-saturated for any $x \in X^\ast $ x element of upper X Superscript asterisk at any $v \in \mathrm {V}(\Gamma )$ v element of normal upper V left parenthesis normal upper Gamma right parenthesis , then $\Gamma $ normal upper Gamma is called $X^\ast $ upper X Superscript asterisk -saturated.

A map $\pi : \Gamma \to \Delta $ pi colon normal upper Gamma right arrow normal upper Delta of two labelled graphs is an immersion if it is injective on the set $\{e \in \mathrm {E}(\Gamma ) \mid \iota (e)=v\}$ StartSet e element of normal upper E left parenthesis normal upper Gamma right parenthesis vertical bar iota left parenthesis e right parenthesis equals v EndSet of all vertices of $\Gamma $ normal upper Gamma . In addition, if $\pi $ pi preserves the labels of edges, then it is called a morphism of labelled graphs. If $\pi $ pi is an injective morphism, then it is called an embedding and we say $\Gamma $ normal upper Gamma embeds in the graph $\Delta $ normal upper Delta .

Remark 2.1. A labelled graph is also called a well-labelled graph (see [Reference Markus-Epstein9, Reference Markus-Epstein10]). Any labelled graph of a free group $F_r$ upper F Subscript r is $F_r$ upper F Subscript r -based.

2.2 Relative Cayley graph

For a group $G=\langle X \mid R \rangle $ upper G equals left angle bracket upper X vertical bar upper R right angle bracket , we denote the Cayley graph of G with respect to the given generating set X by $\mathrm {Cay}(G)$ upper C a y left parenthesis upper G right parenthesis . Recall that $\mathrm {Cay}(G)$ upper C a y left parenthesis upper G right parenthesis is an oriented graph with the vertex set $\mathrm {V}(\mathrm {Cay}(G))=G$ normal upper V left parenthesis upper C a y left parenthesis upper G right parenthesis right parenthesis equals upper G and the edge set $\mathrm {\mathrm {E}}(\mathrm {Cay}(G))= G \times X^\ast $ normal upper E left parenthesis upper C a y left parenthesis upper G right parenthesis right parenthesis equals upper G times upper X Superscript asterisk . An edge $(g,x) \in \mathrm {E}(\mathrm {Cay}(G))$ left parenthesis g comma x right parenthesis element of normal upper E left parenthesis upper C a y left parenthesis upper G right parenthesis right parenthesis originates at g and terminates at $gx$ g x .

For any subgroup $H\leq G$ upper H less than or equals upper G , we denote by $\mathrm {Cay}(G,H)$ upper C a y left parenthesis upper G comma upper H right parenthesis the Cayley graph of G relative to the subgroup H of G, which is an oriented graph with vertex set consisting of all the right cosets of H, $\mathrm {V}(\mathrm {Cay}(G,H)):=\{Hg\mid g\in G\}$ normal upper V left parenthesis upper C a y left parenthesis upper G comma upper H right parenthesis right parenthesis colon equals StartSet upper H g vertical bar g element of upper G EndSet , and edge set $\mathrm {E}(\mathrm {Cay}(G,H))=\mathrm {V}(\mathrm {Cay}(G,H)) \times X^\ast $ normal upper E left parenthesis upper C a y left parenthesis upper G comma upper H right parenthesis right parenthesis equals normal upper V left parenthesis upper C a y left parenthesis upper G comma upper H right parenthesis right parenthesis times upper X Superscript asterisk . We write $\mathrm {Cay}(G,H,H \cdot 1)$ upper C a y left parenthesis upper G comma upper H comma upper H dot 1 right parenthesis when the base point $H \cdot 1$ upper H dot 1 is emphasised. An edge $(Hg,x)$ left parenthesis upper H g comma x right parenthesis originates from the vertex $Hg$ upper H g and ends at $Hgx$ upper H g x . Moreover, a path p originating from $Hg$ upper H g must end at $Hg\mathrm {Lab}(p)$ upper H g upper L a b left parenthesis p right parenthesis . This gives the following result.

Lemma 2.2. Let $H\leq G$ upper H less than or equals upper G be a subgroup of G. A path p originating from the base point $H\cdot 1$ upper H dot 1 in $\mathrm {Cay}(G,H, H \cdot 1)$ upper C a y left parenthesis upper G comma upper H comma upper H dot 1 right parenthesis is a loop if and only if $\mathrm {Lab}(p)\in H$ upper L a b left parenthesis p right parenthesis element of upper H . Hence,

$$ \begin{align*}\mathrm{Lab}(\mathrm{Cay}(G,H), H\cdot 1)=H.\end{align*} $$ upper L a b left parenthesis upper C a y left parenthesis upper G comma upper H right parenthesis comma upper H dot 1 right parenthesis equals upper H period

The following lemma shows that G-based graphs are subgraphs of the relative Cayley graphs.

Lemma 2.3 [Reference Gitik5, Lemma 1.5]

Let $G=\langle X \mid R \rangle $ upper G equals left angle bracket upper X vertical bar upper R right angle bracket be a group and let $(\Gamma ,v_0)$ left parenthesis normal upper Gamma comma v 0 right parenthesis be a graph labelled with X. Denote $\mathrm {Lab}(\Gamma ,v_0)=H$ upper L a b left parenthesis normal upper Gamma comma v 0 right parenthesis equals upper H . Then:

  1. (1) $\Gamma $ normal upper Gamma is G-based if and only if it can be embedded in $\mathrm {Cay}(G,H,H \cdot 1)$ upper C a y left parenthesis upper G comma upper H comma upper H dot 1 right parenthesis ;

  2. (2) $\Gamma $ normal upper Gamma is G-based and $X^\ast $ upper X Superscript asterisk -saturated if and only if it is isomorphic to $\mathrm {Cay}(G,H,H \cdot 1)$ upper C a y left parenthesis upper G comma upper H comma upper H dot 1 right parenthesis .

Note that $\mathrm {Cay}(G)$ upper C a y left parenthesis upper G right parenthesis and $\mathrm {Cay}(G,H)$ upper C a y left parenthesis upper G comma upper H right parenthesis are naturally labelled with the generating set X of G, and there are natural actions of G on $\mathrm {V}(\mathrm {Cay}(G))$ normal upper V left parenthesis upper C a y left parenthesis upper G right parenthesis right parenthesis and $\mathrm {V}(\mathrm {Cay}(G, H))$ normal upper V left parenthesis upper C a y left parenthesis upper G comma upper H right parenthesis right parenthesis given by left translation. The relative Cayley graph $\mathrm {Cay}(G,H)$ upper C a y left parenthesis upper G comma upper H right parenthesis can be defined as the quotient of $\mathrm {Cay}(G)$ upper C a y left parenthesis upper G right parenthesis by the action of H. Moreover, $\mathrm {Cay}(G,H)$ upper C a y left parenthesis upper G comma upper H right parenthesis can be viewed as the 1-skeleton of a covering space of the representation complex of G, so we follow the terminology of Gitik [Reference Gitik5] and refer to the ‘cover’ as the relative Cayley graph.

Scott [Reference Scott13] described the residually finite and LERF properties by the embeddability from subgraphs to covers. Gitik [Reference Gitik5] rephrased Scott’s result as follows.

Theorem 2.4 [Reference Gitik5, Theorem 1.1]

Let $G=\langle X \mid R \rangle $ upper G equals left angle bracket upper X vertical bar upper R right angle bracket be a group:

  1. (1) G is residually finite if and only if for any finite tree $\Gamma $ normal upper Gamma , which is a subgraph of the Cayley graph of G, there exists a finite-index subgroup $H_0$ upper H 0 of G such that $\Gamma $ normal upper Gamma can be embedded in $\mathrm {Cay}(G,H_0)$ upper C a y left parenthesis upper G comma upper H 0 right parenthesis ;

  2. (2) G is LERF if and only if for any finitely generated subgroup H of G and for any finite connected subgraph $(\Gamma ,H \cdot 1)$ left parenthesis normal upper Gamma comma upper H dot 1 right parenthesis of $\mathrm {Cay}(G,H, H \cdot 1)$ upper C a y left parenthesis upper G comma upper H comma upper H dot 1 right parenthesis , there exists a finite-index subgroup $H_0$ upper H 0 of G such that $(\Gamma ,H \cdot 1)$ left parenthesis normal upper Gamma comma upper H dot 1 right parenthesis can be embedded in $\mathrm {Cay}(G,H_0,H_0 \cdot 1)$ upper C a y left parenthesis upper G comma upper H 0 comma upper H 0 dot 1 right parenthesis .

2.3 Precover

The notion of precover was defined by Gitik in [Reference Gitik5] and then actively employed by Markus-Epstein to study algorithmic problems in amalgams of finite groups in [Reference Markus-Epstein9, Reference Markus-Epstein10].

Let $G_1$ upper G 1 and $G_2$ upper G 2 be two groups with generating sets X and Y, respectively, such that $X^\ast \cap Y^\ast = \emptyset $ upper X Superscript asterisk Baseline intersection upper Y Superscript asterisk Baseline equals normal empty set , and let $A=G_1 \mathop {\ast } G_2$ upper A equals upper G 1 asterisk upper G 2 be the free product of $G_1$ upper G 1 and $G_2$ upper G 2 .

Definition 2.5. Let $\Gamma $ normal upper Gamma be a graph labelled with $X \cup Y$ upper X union upper Y .

  1. (1) A subgraph of $\Gamma $ normal upper Gamma is called monochromatic if it is labelled only with X or only with Y. An X-component (respectively Y-component) of $\Gamma $ normal upper Gamma is a maximal connected subgraph of $\Gamma $ normal upper Gamma labelled with X (respectively Y).

  2. (2) A vertex v in $\mathrm {V}(\Gamma )$ normal upper V left parenthesis normal upper Gamma right parenthesis is called X-monochromatic (respectively Y-monochromatic), if all edges connected to v are labelled with X (respectively Y). Otherwise, v is called bichromatic.

  3. (3) We call $\Gamma $ normal upper Gamma a precover of A, if it is A-based, and each X-component of $\Gamma $ normal upper Gamma is a cover of $G_1$ upper G 1 and each Y-component of $\Gamma $ normal upper Gamma is a cover of $G_2$ upper G 2 .

There is an equivalent description of an A-based graph as follows.

Lemma 2.6 [Reference Gitik5]

If $A=G_1 \ast G_2$ upper A equals upper G 1 asterisk upper G 2 is a free product, then any graph labelled with $X \cup Y$ upper X union upper Y is A-based if and only if each $G_i$ upper G Subscript i -component is $G_i$ upper G Subscript i -based for $i=1,2$ i equals 1 comma 2 . In particular, it is A-based if each monochromatic component is a cover of the corresponding free factor.

Lemma 2.7. Let $\Gamma $ normal upper Gamma be a finite, connected, A-based graph labelled with $X \cup Y$ upper X union upper Y and $\Gamma _1$ normal upper Gamma 1 be an X-component. Let P be a loop in $\Gamma $ normal upper Gamma starting from $v_1 \in \Gamma _1$ v 1 element of normal upper Gamma 1 . If $1 \neq \mathrm {Lab}(P) \in G_1$ 1 not equals upper L a b left parenthesis upper P right parenthesis element of upper G 1 , then there is a loop $P'$ upper P prime in $\Gamma _1$ normal upper Gamma 1 starting from $v_1$ v 1 such that $\mathrm {Lab}(P')=\mathrm {Lab}(P)$ upper L a b left parenthesis upper P Superscript prime Baseline right parenthesis equals upper L a b left parenthesis upper P right parenthesis .

Proof. We denote $P = P_1P_2 \cdots P_n$ upper P equals upper P 1 upper P 2 midline horizontal ellipsis upper P Subscript n Baseline , where each $P_j\, (1\leq j\leq n)$ upper P Subscript j Baseline left parenthesis 1 less than or equals j less than or equals n right parenthesis is a maximal connected subpath labelled only with X or Y. Then, $\mathrm {Lab}(P_j) \in G_i$ upper L a b left parenthesis upper P Subscript j Baseline right parenthesis element of upper G Subscript i for some $i\in {1, 2}$ i element of 1 comma 2 . Since $\mathrm {Lab}(P) =\mathrm {Lab}(P_1)\mathrm {Lab}(P_2)\cdots \mathrm {Lab}(P_n) \in G_1$ upper L a b left parenthesis upper P right parenthesis equals upper L a b left parenthesis upper P 1 right parenthesis upper L a b left parenthesis upper P 2 right parenthesis midline horizontal ellipsis upper L a b left parenthesis upper P Subscript n Baseline right parenthesis element of upper G 1 and A is a free product of $G_1$ upper G 1 and $G_2$ upper G 2 , we have $\mathrm {Lab}(P_i)=1$ upper L a b left parenthesis upper P Subscript i Baseline right parenthesis equals 1 for each $P_i$ upper P Subscript i labelled with Y. Then, $P_i$ upper P Subscript i must be closed because $\Gamma $ normal upper Gamma is A-based. Moreover, as $P_i$ upper P Subscript i is maximal, both $P_{i-1}$ upper P Subscript i minus 1 and $P_{i+1}$ upper P Subscript i plus 1 are contained in X-components with $\tau (P_{i-1})=\iota (P_{i})=\tau (P_{i})=\iota (P_{i+1})$ tau left parenthesis upper P Subscript i minus 1 Baseline right parenthesis equals iota left parenthesis upper P Subscript i Baseline right parenthesis equals tau left parenthesis upper P Subscript i Baseline right parenthesis equals iota left parenthesis upper P Subscript i plus 1 Baseline right parenthesis . This implies $P_{i-1}$ upper P Subscript i minus 1 and $P_{i+1}$ upper P Subscript i plus 1 are contained in the same X-component. So we can remove $P_i$ upper P Subscript i and combine $P_{i-1}$ upper P Subscript i minus 1 with $P_{i+1}$ upper P Subscript i plus 1 as a new monochromatic component. After removing all such $P_i$ upper P Subscript i , we obtain a loop $P'$ upper P prime that starts from $v_1$ v 1 and is contained in the X-component $\Gamma _1$ normal upper Gamma 1 such that $\mathrm {Lab}(P')=\mathrm {Lab}(P)$ upper L a b left parenthesis upper P Superscript prime Baseline right parenthesis equals upper L a b left parenthesis upper P right parenthesis .

The precover of free products has a key property.

Lemma 2.8 [Reference Gitik5, Lemma 2.4]

For any groups $G_1=\langle X \mid R \rangle $ upper G 1 equals left angle bracket upper X vertical bar upper R right angle bracket and $G_2=\langle Y \mid T \rangle $ upper G 2 equals left angle bracket upper Y vertical bar upper T right angle bracket , each precover of $G_1 \ast G_2$ upper G 1 asterisk upper G 2 can be embedded in a cover of $G_1\ast G_2$ upper G 1 asterisk upper G 2 with the same set of vertices.

2.4 Pushout and folding

Pushout, considered by Stallings [Reference Stallings14], is an effective way of constructing a new graph from old graphs. For example, in the category of labelled graphs, if $\alpha _1:\Gamma \to \Delta _1$ alpha 1 colon normal upper Gamma right arrow normal upper Delta 1 and $\alpha _2:\Gamma \to \Delta _2$ alpha 2 colon normal upper Gamma right arrow normal upper Delta 2 are injective maps, then the amalgam of $\Delta _1$ normal upper Delta 1 and $\Delta _2$ normal upper Delta 2 denoted by $\Delta _1 \mathop {\ast }\limits _{\Gamma } \Delta _2$ normal upper Delta 1 asterisk Underscript normal upper Gamma Endscripts normal upper Delta 2 is given by the pushout of $\alpha _1$ alpha 1 and $\alpha _2$ alpha 2 :

Commutative diagram showing a pushout: Gamma maps via alpha1 to Delta1 and alpha2 to Delta2. Both Delta1 and Delta2 then map to the amalgamated product Delta1 *_{Gamma} Delta2.

In general, none of the graphs need be connected and none of the maps need be injective. Pushouts do not always exist. The pushout of $\alpha _1:\Gamma \to \Delta _1$ alpha 1 colon normal upper Gamma right arrow normal upper Delta 1 and $\alpha _2:\Gamma \to \Delta _2$ alpha 2 colon normal upper Gamma right arrow normal upper Delta 2 exists if and only if there are orientations of $\Gamma $ normal upper Gamma , $\Delta _1$ normal upper Delta 1 and $\Delta _2$ normal upper Delta 2 which are preserved by $\alpha _1$ alpha 1 and $\alpha _2$ alpha 2 . If we only consider the case in which both $\alpha _1$ alpha 1 and $\alpha _2$ alpha 2 are injective maps, the amalgam can be constructed by performing foldings.

Let $f: \mathrm {E}(\Gamma ) \to X^\ast $ f colon normal upper E left parenthesis normal upper Gamma right parenthesis right arrow upper X Superscript asterisk be a function on the graph $\Gamma $ normal upper Gamma such that $f(\bar {e})=(f(e))^{-1}$ f left parenthesis e overbar right parenthesis equals left parenthesis f left parenthesis e right parenthesis right parenthesis Superscript negative 1 . For any two distinct edges $e_1, e_2\in \mathrm {E}(\Gamma )$ e 1 comma e 2 element of normal upper E left parenthesis normal upper Gamma right parenthesis with $\iota (e_1)=\iota (e_2)$ iota left parenthesis e 1 right parenthesis equals iota left parenthesis e 2 right parenthesis and $e_1 \neq \bar {e}_2$ e 1 not equals e overbar Subscript 2 , if $f(e_1)=f(e_2)$ f left parenthesis e 1 right parenthesis equals f left parenthesis e 2 right parenthesis , then the folding $(e_1,e_2)$ left parenthesis e 1 comma e 2 right parenthesis is the projection of $\Gamma $ normal upper Gamma onto the quotient graph $\Gamma _1$ normal upper Gamma 1 which is constructed by identifying $e_1$ e 1 with $e_2$ e 2 and $\tau (e_1)$ tau left parenthesis e 1 right parenthesis with $\tau (e_2)$ tau left parenthesis e 2 right parenthesis . After performing a finite sequence of foldings as far as possible, the corresponding graph of $\Gamma $ normal upper Gamma is indeed a labelled graph. It can be easily seen that the above amalgam of $\alpha _1$ alpha 1 and $\alpha _2$ alpha 2 can be constructed by taking the disjoint union of graphs, identifying $\alpha _1(\Gamma )$ alpha 1 left parenthesis normal upper Gamma right parenthesis with $\alpha _2(\Gamma )$ alpha 2 left parenthesis normal upper Gamma right parenthesis and then performing a finite sequence of foldings until a labelled graph is obtained.

3 Wilton’s proof for free groups

In this section, we recall the specific structure of Wilton’s proof that any finitely generated infinite-index subgroup of a free group with rank greater than one is $\mathcal A$ script upper A -separable.

3.1 Jordan’s theorem

An action of a group H on $\{1,2,\ldots ,n\}$ StartSet 1 comma 2 comma ellipsis comma n EndSet is called symmetric or alternating according to whether the image of $H\to S_n$ upper H right arrow upper S Subscript n is $S_n$ upper S Subscript n or $A_n$ upper A Subscript n . A subgroup $G \leq S_n$ upper G less than or equals upper S Subscript n is called transitive if it acts transitively on $\{1,2,\ldots ,n\}$ StartSet 1 comma 2 comma ellipsis comma n EndSet , and it is called primitive if, in addition, it does not preserve any nontrivial partition.

The main tool for proving Lemma 3.3 is Jordan’s theorem, which gives a criterion for an action to be symmetric or alternating.

Theorem 3.1 (Jordan’s theorem [Reference Dixon and Mortimer4])

For each $k> 2$ k greater than 2 , there exists N such that if $n> N$ n greater than upper N , $G \leq S_n$ upper G less than or equals upper S Subscript n is a primitive subgroup and there exists $\gamma \in G \setminus \{e\} $ gamma element of upper G minus StartSet e EndSet which moves fewer than k elements, then $G=S_n$ upper G equals upper S Subscript n or $A_n$ upper A Subscript n .

Note that if $G \leq S_n$ upper G less than or equals upper S Subscript n is transitive, then the cardinality of any partition preserved by G divides n. Therefore, if p is prime, then $G\leq S_p$ upper G less than or equals upper S Subscript p is primitive if and only if G is transitive. The next result is an immediate corollary of Theorem 3.1.

Lemma 3.2. For each $k> 2$ k greater than 2 , there exists N such that for any prime $p>N$ p greater than upper N and any transitive subgroup $G \leq S_p$ upper G less than or equals upper S Subscript p , if there exists $\gamma \in G \setminus \{e\}$ gamma element of upper G minus StartSet e EndSet which moves fewer than k elements, then $G=S_p$ upper G equals upper S Subscript p or $A_p$ upper A Subscript p .

3.2 Finite covers and Wilton’s construction

Let $F_r$ upper F Subscript r be a free group of rank $r>1$ r greater than 1 with the generating set $X=\{x_1,\ldots ,x_r\}$ upper X equals StartSet x 1 comma ellipsis comma x Subscript r Baseline EndSet . Stallings [Reference Stallings14] showed that any finite graph labelled with X can be embedded into a cover without increasing the number of vertices. Wilton made a modification in the construction to force the action of $F_r$ upper F Subscript r on the vertex set of the covering to be alternating or symmetric (see [Reference Wilton18, Lemmas 18 and 19]). We summarise these results in the following lemma and describe Wilton’s specific construction in detail in the proof. We use the same approach in our proof of Theorem 1.5.

Lemma 3.3. Let Z be a finite, connected graph labelled with X. If Z is not $X^\ast $ upper X Superscript asterisk -saturated, then there is a finite cover Y of $F_r$ upper F Subscript r such that Z embeds in Y and the action of $F_r$ upper F Subscript r on $\mathrm {V}(Y)$ normal upper V left parenthesis upper Y right parenthesis is alternating or symmetric.

Proof. Suppose Z has k vertices. By Lemma 3.2, there exists a sufficiently large prime p for $k+5$ k plus 5 such that for any transitive subgroup $G\leq S_p$ upper G less than or equals upper S Subscript p , if there exists ${\gamma \in G \setminus \{e\} }$ gamma element of upper G minus StartSet e EndSet which moves fewer than $k+5$ k plus 5 elements, then $G=S_p$ upper G equals upper S Subscript p or $A_p$ upper A Subscript p . We shall construct a finite cover Y with p vertices such that $F_r$ upper F Subscript r acts on $\mathrm {V}(Y)$ normal upper V left parenthesis upper Y right parenthesis transitively and there is an element of $F_r$ upper F Subscript r moving fewer than $k+5$ k plus 5 vertices.

Since Z is not an $X^\ast $ upper X Superscript asterisk -saturated graph, there are (not necessarily distinct) vertices $a,b$ a comma b and some $x \in X$ x element of upper X such that Z is not x-saturated at a and not $x^{-1}$ x Superscript negative 1 -saturated at b. Without loss of generality, we take $x=x_1$ x equals x 1 .

We begin by constructing two particular labelled graphs $W_n$ upper W Subscript n and $V_s$ upper V Subscript s with ${n=p-k-4}$ n equals p minus k minus 4 and $4$ 4 vertices, respectively, such that $x_i\in X$ x Subscript i Baseline element of upper X acts trivially on $\mathrm {V}(W_n)$ normal upper V left parenthesis upper W Subscript n Baseline right parenthesis for each $i \geq 2$ i greater than or equals 2 and $x_2$ x 2 acts nontrivially on $\mathrm {V}(V_s)$ normal upper V left parenthesis normal upper V Subscript s Baseline right parenthesis .

The construction of $W_n$ upper W Subscript n is as follows. We first take an interval with n vertices, denoted by $w_1,\ldots ,w_n$ w 1 comma ellipsis comma w Subscript n Baseline and denote the $n-1$ n minus 1 edges by $e_1,\ldots ,e_{n-1}$ e 1 comma ellipsis comma e Subscript n minus 1 Baseline . We label all edges with $x_1$ x 1 and orient these edges by

$$ \begin{align*}\iota(e_1)=w_1, \quad \tau(e_j)=\iota(e_{j+1})=w_{j+1}, \quad \tau(e_{n-1})=w_n.\end{align*} $$ StartLayout 1st Row iota left parenthesis e 1 right parenthesis equals w 1 comma tau left parenthesis e Subscript j Baseline right parenthesis equals iota left parenthesis e Subscript j plus 1 Baseline right parenthesis equals w Subscript j plus 1 Baseline comma tau left parenthesis e Subscript n minus 1 Baseline right parenthesis equals w Subscript n Baseline period EndLayout

Then, for each generator $x_i \in X\, (i \geq 2)$ x Subscript i Baseline element of upper X left parenthesis i greater than or equals 2 right parenthesis and each vertex $w_j\, (1 \leq j \leq n)$ w Subscript j Baseline left parenthesis 1 less than or equals j less than or equals n right parenthesis , we attach an oriented edge labelled with $x_i$ x Subscript i from $w_j$ w Subscript j to itself. The resulting labelled graph is $W_n$ upper W Subscript n . (See the left-hand side of Figure 1.) For each $i \geq 2$ i greater than or equals 2 , $x_i$ x Subscript i fixes all vertices of $W_n$ upper W Subscript n and, hence, acts trivially on $\mathrm {V}(W_n)$ normal upper V left parenthesis upper W Subscript n Baseline right parenthesis . Note that the unsaturated vertices in $W_n$ upper W Subscript n are $w_1$ w 1 and $w_n$ w Subscript n ; $w_1$ w 1 is not $x_1^{-1}$ x 1 Superscript negative 1 -saturated and $w_n$ w Subscript n is not $x_1$ x 1 -saturated.

Figure 1

The graph $W_4$ upper W 4 on the left and $V_{+1,-1}$ upper V Subscript plus 1 comma negative 1 on the right.

Two directed graphs side by side, each with labeled nodes and arrows showing different edge and loop structures for x one and x two. See long description.
Figure 1 Long description

The left panel shows a directed graph with four nodes labeled w one to w four arranged in a horizontal line. Each node has a loop labeled x two. Arrows labeled x one connect w one to w two, w two to w three, and w three to w four. The right panel shows a directed graph with four nodes labeled v one to v four arranged in a square. v one connects to v two with an arrow labeled x two, v two connects to v four with x two, v four connects to v three with x two, and v three connects to v one with x two. There are loops labeled x one at v three and v four, and an arrow labeled x two from v one to v three. An arrow labeled x one connects v one to v two.

The construction of $V_s$ upper V Subscript s is as follows. Here, s denotes an r-tuple ${(s_i) {\kern-1pt}\in{\kern-1pt} \{\pm 1\}\, (1 \leq i \leq r)}$ left parenthesis s Subscript i Baseline right parenthesis element of StartSet plus or minus 1 EndSet left parenthesis 1 less than or equals i less than or equals r right parenthesis . We first take four vertices, denoted by $v_1,v_2,v_3,v_4$ v 1 comma v 2 comma v 3 comma v 4 and attach an edge labelled with $x_1$ x 1 from $v_1$ v 1 to $v_2$ v 2 . For each $i>2$ i greater than 2 , we attach a loop labelled with $x_i$ x Subscript i to both $v_1$ v 1 and $v_2$ v 2 . For each $i \neq 2$ i not equals 2 , if $s_i=1$ s Subscript i Baseline equals 1 , we attach a loop labelled with $x_i$ x Subscript i to both $v_3$ v 3 and $v_4$ v 4 ; if $s_i=-1$ s Subscript i Baseline equals negative 1 , we attach two edges both labelled with $x_i$ x Subscript i , from $v_3$ v 3 to $v_4$ v 4 and from $v_4$ v 4 to $v_3$ v 3 , respectively. If $s_2=+1$ s 2 equals plus 1 , we attach four edges labelled with $x_2$ x 2 , from $v_1$ v 1 to $v_3$ v 3 , from $v_3$ v 3 to $v_1$ v 1 , from $v_2$ v 2 to $v_4$ v 4 and from $v_4$ v 4 to $v_2$ v 2 , respectively; if $s_2=-1$ s 2 equals negative 1 , we attach a loop of length four with vertices $v_1,v_2,v_3$ v 1 comma v 2 comma v 3 and $v_4$ v 4 , each edge is labelled with $x_2$ x 2 . (See the right-hand side of Figure 1.) The resulting labelled graph is $V_s$ upper V Subscript s and $x_2$ x 2 acts nontrivially on $\mathrm {V}(V_s)$ normal upper V left parenthesis normal upper V Subscript s Baseline right parenthesis . Note that the unsaturated vertices in $V_s$ upper V Subscript s are $v_1$ v 1 and $v_2$ v 2 ; $v_1$ v 1 is not $x_1^{-1}$ x 1 Superscript negative 1 -saturated and $v_2$ v 2 is not $x_1$ x 1 -saturated.

We now construct the desired Y by attaching some edges to $Z\cup W_n\cup V_s$ upper Z union upper W Subscript n Baseline union upper V Subscript s as follows. First, for the six unsaturated vertices a, b in Z, $w_1$ w 1 , $w_{n}$ w Subscript n in $W_{n}$ upper W Subscript n and $v_1$ v 1 , $v_2$ v 2 in $V_s$ upper V Subscript s , we attach three edges labelled with $x_1$ x 1 : one from a to $w_1$ w 1 , one from $w_{n}$ w Subscript n to $v_1$ v 1 and one from $v_2$ v 2 to b. The resulting graph $Y'$ upper Y prime has $k+n+4=p$ k plus n plus 4 equals p vertices and it is a labelled graph. Finally, by adding some edges to $Y'$ upper Y prime , we obtain the finite cover Y without adding new vertices:

$$ \begin{align*}|\mathrm{V}(Y)|=|\mathrm{V}(Y')|=p.\end{align*} $$ StartLayout 1st Row StartAbsoluteValue normal upper V left parenthesis upper Y right parenthesis EndAbsoluteValue equals StartAbsoluteValue normal upper V left parenthesis upper Y Superscript prime Baseline right parenthesis EndAbsoluteValue equals p period EndLayout

Note that Y is connected and the vertices $w_1,\ldots ,w_{n}$ w 1 comma ellipsis comma w Subscript n Baseline in Y are fixed by the action of $x_2$ x 2 . It follows that $F_r$ upper F Subscript r acts on $\mathrm {V}(Y)$ normal upper V left parenthesis upper Y right parenthesis transitively and $x_2$ x 2 moves fewer than $p-n+1=k+5$ p minus n plus 1 equals k plus 5 vertices. Therefore, Y meets the requirements specified in the first paragraph and the proof is complete.

Remark 3.4. To connect $W_n$ upper W Subscript n to Z, we require the existence of an unsaturated vertex on Z, meaning that Z is not a cover of $F_r$ upper F Subscript r . If there are multiple monochromatic components, we correspondingly require that a certain $F_r$ upper F Subscript r -component is not a cover.

4 Kurosh decomposition structure derived from the precover

In this section, we introduce the Markus-Epstein algorithm (see [Reference Markus-Epstein10]), which provides a method for constructing Kurosh decompositions of subgroups via precovers. We employ this approach to characterise the structure of subgroups defined by labelled graphs.

Theorem 4.1 (Kurosh subgroup theorem [Reference Kurosch7])

Suppose $G = \ast _{i \in I} G_i$ upper G equals asterisk Underscript i element of upper I Endscripts upper G Subscript i is a free product and H is a subgroup of G. Then, $H\cong (\ast H_j)\ast F$ upper H approximately equals left parenthesis asterisk upper H Subscript j Baseline right parenthesis asterisk upper F , where each $H_j$ upper H Subscript j is isomorphic to an intersection $H \cap G^{k_i}_i$ upper H intersection upper G Subscript i Superscript k Super Subscript i of H with a conjugate of some $G_i$ upper G Subscript i . Further, the set $\{H_j\}$ StartSet upper H Subscript j Baseline EndSet is unique up to conjugation and re-indexing, and the rank of the free group F is uniquely determined.

Let $G=G_1 \ast G_2$ upper G equals upper G 1 asterisk upper G 2 be the free product of $G_1$ upper G 1 and $G_2$ upper G 2 with generating sets X and Y, respectively. Let $\Gamma $ normal upper Gamma be a finite graph labelled with $X\cup Y$ upper X union upper Y and $v_0$ v 0 be the base point of $\Gamma $ normal upper Gamma . Furthermore, we assume that $\Gamma $ normal upper Gamma is connected, G-based and denote $\mathrm {Lab}(\Gamma ,v_0)= H \leq G$ upper L a b left parenthesis normal upper Gamma comma v 0 right parenthesis equals upper H less than or equals upper G .

For a monochromatic component C of $\Gamma $ normal upper Gamma with base point $v \in \mathrm {V}(C)$ v element of normal upper V left parenthesis upper C right parenthesis , let $T(C)$ upper T left parenthesis upper C right parenthesis be a spanning tree of C with root vertex v. Since $\Gamma $ normal upper Gamma is connected, there is an approach path $P_v$ upper P Subscript v from $v_0$ v 0 to v, which we denote $\mathrm {Lab}(P_v) \equiv g_v$ upper L a b left parenthesis upper P Subscript v Baseline right parenthesis identical to g Subscript v . Without loss of generality, we can assume that $P_v \cap C=\{v\}$ upper P Subscript v Baseline intersection upper C equals StartSet v EndSet . (Otherwise, since C is connected, we can assume the subpath $P_c=P_v \cap C$ upper P Subscript c Baseline equals upper P Subscript v Baseline intersection upper C is connected, and then change v to be the initial point of $P_c$ upper P Subscript c and remove $P_c$ upper P Subscript c from $P_v$ upper P Subscript v .) If $v_0 \in \mathrm {V}(C)$ v 0 element of normal upper V left parenthesis upper C right parenthesis , we choose $v=v_0$ v equals v 0 . Let $\Gamma '$ normal upper Gamma prime be a connected subgraph of $\Gamma $ normal upper Gamma obtained by removing edges in $\mathrm {E}(C)\setminus \mathrm {E}(T(C))$ normal upper E left parenthesis upper C right parenthesis minus normal upper E left parenthesis upper T left parenthesis upper C right parenthesis right parenthesis . More precisely,

$$ \begin{align*}\mathrm{V}(\Gamma')=\mathrm{V}(\Gamma) \quad\text{and}\quad \mathrm{E}(\Gamma')=(\mathrm{E}(\Gamma)\setminus \mathrm{E}(C)) \cup \mathrm{E}(T(C)).\end{align*} $$ StartLayout 1st Row normal upper V left parenthesis normal upper Gamma Superscript prime Baseline right parenthesis equals normal upper V left parenthesis normal upper Gamma right parenthesis and normal upper E left parenthesis normal upper Gamma Superscript prime Baseline right parenthesis equals left parenthesis normal upper E left parenthesis normal upper Gamma right parenthesis minus normal upper E left parenthesis upper C right parenthesis right parenthesis union normal upper E left parenthesis upper T left parenthesis upper C right parenthesis right parenthesis period EndLayout

Following these assumptions, Markus-Epstein’s algorithm is based on the following result [Reference Markus-Epstein10, Lemma 4.2].

Lemma 4.2. We have $H{\kern-1pt}={\kern-1pt}\langle g_v \mathrm {Lab}(C,v) g_v^{-1}, \mathrm {Lab}(\Gamma ', v_0)\rangle {\kern-1pt}={\kern-1pt}g_v \mathrm {Lab}(C,v) g_v^{-1}{\kern-1pt}\ast{\kern-1pt} \mathrm {Lab}(\Gamma ', v_0)$ upper H equals left angle bracket g Subscript v Baseline upper L a b left parenthesis upper C comma v right parenthesis g Subscript v Superscript negative 1 Baseline comma upper L a b left parenthesis normal upper Gamma prime comma v 0 right parenthesis right angle bracket equals g Subscript v Baseline upper L a b left parenthesis upper C comma v right parenthesis g Subscript v Superscript negative 1 Baseline asterisk upper L a b left parenthesis normal upper Gamma prime comma v 0 right parenthesis .

Lemma 4.2 shows that every monochromatic component C with nontrivial $\mathrm {Lab}(C,v)$ upper L a b left parenthesis upper C comma v right parenthesis corresponds to a free factor $g_v \mathrm {Lab}(C,v) g_v^{-1}$ g Subscript v Baseline upper L a b left parenthesis upper C comma v right parenthesis g Subscript v Superscript negative 1 of $\mathrm {Lab}(\Gamma ,v_0)$ upper L a b left parenthesis normal upper Gamma comma v 0 right parenthesis . This gives the following free decomposition of $\mathrm {Lab}(\Gamma ,v_0)$ upper L a b left parenthesis normal upper Gamma comma v 0 right parenthesis .

Lemma 4.3. Let $\Gamma $ normal upper Gamma be a finite, connected, G-based graph labelled with $X\cup Y$ upper X union upper Y . If there are finitely many monochromatic components $C_i\, (1 \leq i \leq k)$ upper C Subscript i Baseline left parenthesis 1 less than or equals i less than or equals k right parenthesis which are not trees, then

$$ \begin{align*}\mathrm{Lab}(\Gamma,v_0)=(\ast_{1\leq i\leq k}\mathrm{Lab}(P_{i}) \mathrm{Lab}(C_i,v_i) \mathrm{Lab}(P_{i})^{-1}) \ast \mathrm{Lab}(\Delta,v_0),\end{align*} $$ StartLayout 1st Row upper L a b left parenthesis normal upper Gamma comma v 0 right parenthesis equals left parenthesis asterisk Underscript 1 less than or equals i less than or equals k Endscripts upper L a b left parenthesis upper P Subscript i Baseline right parenthesis upper L a b left parenthesis upper C Subscript i Baseline comma v Subscript i Baseline right parenthesis upper L a b left parenthesis upper P Subscript i Baseline right parenthesis Superscript negative 1 Baseline right parenthesis asterisk upper L a b left parenthesis normal upper Delta comma v 0 right parenthesis comma EndLayout

where $P_{i}$ upper P Subscript i is an approach path from the base point $v_0$ v 0 of $\Gamma $ normal upper Gamma to the base point $v_i$ v Subscript i of $C_i$ upper C Subscript i , the subgraph $\Delta $ normal upper Delta is obtained from $\Gamma $ normal upper Gamma by removing all edges in each component $C_i$ upper C Subscript i that do not belong to its spanning tree $T(C_i)$ upper T left parenthesis upper C Subscript i Baseline right parenthesis and $\mathrm {Lab}(\Delta ,v_0)$ upper L a b left parenthesis normal upper Delta comma v 0 right parenthesis is a free group that does not conjugate into $G_1$ upper G 1 or $G_2$ upper G 2 .

Proof. Let $\mathrm {Lab}(P_{i})=g_{i}$ upper L a b left parenthesis upper P Subscript i Baseline right parenthesis equals g Subscript i . We proceed by induction on the number k of monochromatic components which are not trees. Since there are finitely many such monochromatic components, the process is finite. When $k=1$ k equals 1 , by Lemma 4.2,

$$ \begin{align*}\mathrm{Lab}(\Gamma,v_0)=g_{1} \mathrm{Lab}(C_1,v_1) g_{1}^{-1}\ast \mathrm{Lab}(\Gamma', v_0),\end{align*} $$ StartLayout 1st Row upper L a b left parenthesis normal upper Gamma comma v 0 right parenthesis equals g 1 upper L a b left parenthesis upper C 1 comma v 1 right parenthesis g 1 Superscript negative 1 Baseline asterisk upper L a b left parenthesis normal upper Gamma prime comma v 0 right parenthesis comma EndLayout

so that $\Gamma '$ normal upper Gamma prime is the required $\Delta $ normal upper Delta .

When $k>1$ k greater than 1 , we start from the monochromatic component $C_1$ upper C 1 containing the base point $v_0$ v 0 of $\Gamma $ normal upper Gamma such that $v_1=v_0 \in \mathrm {V}(C_1)$ v 1 equals v 0 element of normal upper V left parenthesis upper C 1 right parenthesis . By Lemma 4.2,

$$ \begin{align*}\mathrm{Lab}(\Gamma,v_0)=g_{1} \mathrm{Lab}(C_1,v_1) g_{1}^{-1}\ast \mathrm{Lab}(\Gamma", v_0).\end{align*} $$ StartLayout 1st Row upper L a b left parenthesis normal upper Gamma comma v 0 right parenthesis equals g 1 upper L a b left parenthesis upper C 1 comma v 1 right parenthesis g 1 Superscript negative 1 Baseline asterisk upper L a b left parenthesis normal upper Gamma quotation mark comma v 0 right parenthesis period EndLayout

Since $\Gamma "$ normal upper Gamma quotation mark embeds in $\Gamma $ normal upper Gamma and has fewer monochromatic components which are not trees than $\Gamma $ normal upper Gamma , by the induction hypothesis,

$$ \begin{align*}\mathrm{Lab}(\Gamma",v_0)=(\ast_{2\leq i\leq k} g_i \mathrm{Lab}(C_i,v_i) g_i^{-1}) \ast \mathrm{Lab}(\Delta,v_0).\end{align*} $$ StartLayout 1st Row upper L a b left parenthesis normal upper Gamma quotation mark comma v 0 right parenthesis equals left parenthesis asterisk Underscript 2 less than or equals i less than or equals k Endscripts g Subscript i Baseline upper L a b left parenthesis upper C Subscript i Baseline comma v Subscript i Baseline right parenthesis g Subscript i Superscript negative 1 Baseline right parenthesis asterisk upper L a b left parenthesis normal upper Delta comma v 0 right parenthesis period EndLayout

By [Reference Markus-Epstein10, Lemma 5.2], $\mathrm {Lab}(\Delta ,v_0)$ upper L a b left parenthesis normal upper Delta comma v 0 right parenthesis is a free group that does not conjugate into $G_1$ upper G 1 or $G_2$ upper G 2 . Combining the above two equations completes the proof.

The free decomposition in Lemma 4.3 coincides with the standard Kurosh decomposition.

Lemma 4.4. In Lemma 4.3, each free factor

$$ \begin{align*}L_i:=\mathrm{Lab}(P_{i}) \mathrm{Lab}(C_i,v_i) \mathrm{Lab}(P_{i})^{-1}=\mathrm{Lab}(\Gamma,v_0) \cap G_k^{\mathrm{Lab}(P_{i})}\end{align*} $$ StartLayout 1st Row upper L Subscript i Baseline colon equals upper L a b left parenthesis upper P Subscript i Baseline right parenthesis upper L a b left parenthesis upper C Subscript i Baseline comma v Subscript i Baseline right parenthesis upper L a b left parenthesis upper P Subscript i Baseline right parenthesis Superscript negative 1 Baseline equals upper L a b left parenthesis normal upper Gamma comma v 0 right parenthesis intersection upper G Subscript k Superscript upper L a b left parenthesis upper P Super Subscript i Superscript right parenthesis EndLayout

for some $k \in \{1,2\}$ k element of StartSet 1 comma 2 EndSet and $\mathrm {Lab}(C_i,v_i) \leq G_k$ upper L a b left parenthesis upper C Subscript i Baseline comma v Subscript i Baseline right parenthesis less than or equals upper G Subscript k .

Proof. Since $C_i$ upper C Subscript i is a monochromatic component, $\mathrm {Lab}(C_i,v_i)\leq G_k$ upper L a b left parenthesis upper C Subscript i Baseline comma v Subscript i Baseline right parenthesis less than or equals upper G Subscript k for some $k \in \{1,2\}$ k element of StartSet 1 comma 2 EndSet and clearly $L_i \leq \mathrm {Lab}(\Gamma ,v_0) \cap G_k^{\mathrm {Lab}(P_i)}$ upper L Subscript i Baseline less than or equals upper L a b left parenthesis normal upper Gamma comma v 0 right parenthesis intersection upper G Subscript k Superscript upper L a b left parenthesis upper P Super Subscript i Superscript right parenthesis . Next, we show $ \mathrm {Lab}(\Gamma ,v_0) \cap G_k^{\mathrm {Lab}(P_i)}\leq L_i.$ upper L a b left parenthesis normal upper Gamma comma v 0 right parenthesis intersection upper G Subscript k Superscript upper L a b left parenthesis upper P Super Subscript i Superscript right parenthesis Baseline less than or equals upper L Subscript i Baseline period

For any $\gamma \in \mathrm {Lab}(\Gamma ,v_0)\cap G_k^{\mathrm {Lab}(P_i)}$ gamma element of upper L a b left parenthesis normal upper Gamma comma v 0 right parenthesis intersection upper G Subscript k Superscript upper L a b left parenthesis upper P Super Subscript i Superscript right parenthesis , we can denote $\gamma =\mathrm {Lab}(P_i) g \mathrm {Lab}(P_i)^{-1}$ gamma equals upper L a b left parenthesis upper P Subscript i Baseline right parenthesis g upper L a b left parenthesis upper P Subscript i Baseline right parenthesis Superscript negative 1 for some $g\in G_k$ g element of upper G Subscript k . Since $\gamma \in \mathrm {Lab}(\Gamma ,v_0)$ gamma element of upper L a b left parenthesis normal upper Gamma comma v 0 right parenthesis , there is a loop $P_\gamma $ upper P Subscript gamma starting from $v_0$ v 0 with $\gamma = \mathrm {Lab}(P_{\gamma })$ gamma equals upper L a b left parenthesis upper P Subscript gamma Baseline right parenthesis . Thus, $g = \mathrm {Lab}(P_i)^{-1} \mathrm {Lab}(P_{\gamma })\mathrm {Lab}(P_i)=\mathrm {Lab}(\bar {P}_iP_{\gamma }P_i) \in G_k$ g equals upper L a b left parenthesis upper P Subscript i Baseline right parenthesis Superscript negative 1 Baseline upper L a b left parenthesis upper P Subscript gamma Baseline right parenthesis upper L a b left parenthesis upper P Subscript i Baseline right parenthesis equals upper L a b left parenthesis upper P overbar Subscript i Baseline upper P Subscript gamma Baseline upper P Subscript i Baseline right parenthesis element of upper G Subscript k , where $\bar {P}_iP_{\gamma }P_i$ upper P overbar Subscript i Baseline upper P Subscript gamma Baseline upper P Subscript i is a loop starting from $v_i$ v Subscript i . By Lemma 2.7, there is a loop $P_{g} \subset C_i$ upper P Subscript g Baseline subset of upper C Subscript i starting from $v_i$ v Subscript i such that

$$ \begin{align*}g=\mathrm{Lab}(\bar{P}_iP_{\gamma}P_i)=\mathrm{Lab}(P_{g}).\end{align*} $$ StartLayout 1st Row g equals upper L a b left parenthesis upper P overbar Subscript i Baseline upper P Subscript gamma Baseline upper P Subscript i Baseline right parenthesis equals upper L a b left parenthesis upper P Subscript g Baseline right parenthesis period EndLayout

This implies that $g \in \mathrm {Lab}(C_i,v_i)$ g element of upper L a b left parenthesis upper C Subscript i Baseline comma v Subscript i Baseline right parenthesis and then $\gamma \in L_i$ gamma element of upper L Subscript i .

5 Proof of the main theorem

In this section, we employ Wilton’s construction and Markus-Epstein’s algorithm to establish Theorem 1.5.

Let $F_r=\langle X \mid -\rangle $ upper F Subscript r Baseline equals left angle bracket upper X vertical bar minus right angle bracket be a free group of rank $r>1$ r greater than 1 , $G=\langle Y \mid R \rangle $ upper G equals left angle bracket upper Y vertical bar upper R right angle bracket a group and

$$ \begin{align*}H= \langle h_1, h_2,\ldots,h_n\rangle \leq F_r \ast G.\end{align*} $$ upper H equals left angle bracket h 1 comma h 2 comma ellipsis comma h Subscript n Baseline right angle bracket less than or equals upper F Subscript r Baseline asterisk upper G period

For any finite set $\{\gamma _1,\ldots ,\gamma _m\} \subset (F_r\ast G)\setminus H$ StartSet gamma 1 comma ellipsis comma gamma Subscript m Baseline EndSet subset of left parenthesis upper F Subscript r Baseline asterisk upper G right parenthesis minus upper H , we pick a finite connected subgraph $\Gamma $ normal upper Gamma of $\mathrm {Cay}(F_r \ast G,H)$ upper C a y left parenthesis upper F Subscript r Baseline asterisk upper G comma upper H right parenthesis ,

(5.1) $$ \begin{align} \Gamma=\bigcup_{i=1}^n p_i\bigcup_{j=1}^m p^{\prime}_j \subset \mathrm{Cay}(F_r \ast G,H), \end{align} $$ normal upper Gamma equals union Underscript i equals 1 Overscript n Endscripts p Subscript i Baseline union Underscript j equals 1 Overscript m Endscripts p Subscript j Superscript prime Baseline subset of upper C a y left parenthesis upper F Subscript r Baseline asterisk upper G comma upper H right parenthesis comma

where all the loops $p_i$ p Subscript i and paths $p^{\prime }_j$ p prime Subscript j have the same initial point $H \cdot 1$ upper H dot 1 , and the labels of $p_i$ p Subscript i and $p^{\prime }_j$ p prime Subscript j are given by $h_i$ h Subscript i and $\gamma _j$ gamma Subscript j , respectively. Note that $\Gamma $ normal upper Gamma is an $F_r\ast G$ upper F Subscript r Baseline asterisk upper G -based labelled graph because it embeds in the $F_r\ast G$ upper F Subscript r Baseline asterisk upper G -based graph $\mathrm {Cay}(F_r \ast G,H)$ upper C a y left parenthesis upper F Subscript r Baseline asterisk upper G comma upper H right parenthesis . Moreover, any $F_r$ upper F Subscript r -component of $\Gamma $ normal upper Gamma is $F_r$ upper F Subscript r -based and any G-component of $\Gamma $ normal upper Gamma is G-based by Lemma 2.6.

Lemma 5.1. We have $\mathrm {Lab}(\Gamma , H \cdot 1 )=\mathrm {Lab}(\mathrm {Cay}(F_r \ast G,H), H \cdot 1)=H$ upper L a b left parenthesis normal upper Gamma comma upper H dot 1 right parenthesis equals upper L a b left parenthesis upper C a y left parenthesis upper F Subscript r Baseline asterisk upper G comma upper H right parenthesis comma upper H dot 1 right parenthesis equals upper H .

Proof. The last equality follows from Lemma 2.2. Since $\Gamma $ normal upper Gamma is the subgraph of $\mathrm {Cay}(F_r \ast G,H)$ upper C a y left parenthesis upper F Subscript r Baseline asterisk upper G comma upper H right parenthesis , each loop in $\mathrm {Lab}(\Gamma , H \cdot 1 )$ upper L a b left parenthesis normal upper Gamma comma upper H dot 1 right parenthesis is again a loop in ${\mathrm {Lab}(\mathrm {Cay}(F_r \ast G,H), H \cdot 1)}$ upper L a b left parenthesis upper C a y left parenthesis upper F Subscript r Baseline asterisk upper G comma upper H right parenthesis comma upper H dot 1 right parenthesis , that is, $\mathrm {Lab}(\Gamma , H \cdot 1 ) \leq \mathrm {Lab}(\mathrm {Cay}(F_r \ast G,H), H \cdot 1)$ upper L a b left parenthesis normal upper Gamma comma upper H dot 1 right parenthesis less than or equals upper L a b left parenthesis upper C a y left parenthesis upper F Subscript r Baseline asterisk upper G comma upper H right parenthesis comma upper H dot 1 right parenthesis . Conversely, since the labels of the loops $p_i\ (1 \leq i \leq n)$ p Subscript i Baseline left parenthesis 1 less than or equals i less than or equals n right parenthesis are given by the generators $h_i$ h Subscript i of H, we clearly have $H \leq \mathrm {Lab}(\Gamma , H \cdot 1)$ upper H less than or equals upper L a b left parenthesis normal upper Gamma comma upper H dot 1 right parenthesis .

We can describe the $\mathcal A \cup \mathcal S$ script upper A union script upper S -separability of H via the embedding property of $\Gamma $ normal upper Gamma .

Proposition 5.2. If there exists a subgroup $H_0\leq F_r \ast G$ upper H 0 less than or equals upper F Subscript r Baseline asterisk upper G with prime index p for $k=|V(\Gamma )|+2$ k equals StartAbsoluteValue upper V left parenthesis normal upper Gamma right parenthesis EndAbsoluteValue plus 2 as in Lemma 3.2, such that $(\Gamma , H\cdot 1)$ left parenthesis normal upper Gamma comma upper H dot 1 right parenthesis can be embedded in ${\mathrm {Cay}(F_r \ast G,H_0,H_0\cdot 1)}$ upper C a y left parenthesis upper F Subscript r Baseline asterisk upper G comma upper H 0 comma upper H 0 dot 1 right parenthesis , and there is a $\gamma \in F_r \ast G$ gamma element of upper F Subscript r Baseline asterisk upper G acting on $\mathrm {V}(\mathrm {Cay}(F_r \ast G,H_0))$ normal upper V left parenthesis upper C a y left parenthesis upper F Subscript r Baseline asterisk upper G comma upper H 0 right parenthesis right parenthesis nontrivially and moving fewer than k vertices, then H is $\mathcal A \cup \mathcal S$ script upper A union script upper S -separable in $F_r \ast G$ upper F Subscript r Baseline asterisk upper G .

Proof. Following the notation above, we show that there exists a surjection f from $F_r \ast G$ upper F Subscript r Baseline asterisk upper G to an alternating or symmetric group such that $f(\gamma _j) \notin f(H)$ f left parenthesis gamma Subscript j Baseline right parenthesis not an element of f left parenthesis upper H right parenthesis for $1 \leq j \leq m$ 1 less than or equals j less than or equals m .

By the hypothesis, there exists a subgroup $H_0\leq F_r \ast G$ upper H 0 less than or equals upper F Subscript r Baseline asterisk upper G with prime index p as in Lemma 3.2 such that $(\Gamma , H\cdot 1)$ left parenthesis normal upper Gamma comma upper H dot 1 right parenthesis embeds in $\mathrm {Cay}(F_r \ast G,H_0,H_0\cdot 1)$ upper C a y left parenthesis upper F Subscript r Baseline asterisk upper G comma upper H 0 comma upper H 0 dot 1 right parenthesis . Note that

$$ \begin{align*}|\mathrm{V}(\mathrm{Cay}(F_r \ast G,H_0))|=p, \end{align*} $$ StartAbsoluteValue normal upper V left parenthesis upper C a y left parenthesis upper F Subscript r Baseline asterisk upper G comma upper H 0 right parenthesis right parenthesis EndAbsoluteValue equals p comma

the action of $F_r \ast G$ upper F Subscript r Baseline asterisk upper G on $\mathrm {V}(\mathrm {Cay}(F_r \ast G,H_0))$ normal upper V left parenthesis upper C a y left parenthesis upper F Subscript r Baseline asterisk upper G comma upper H 0 right parenthesis right parenthesis is transitive and there exists $\gamma \in F_r \ast G$ gamma element of upper F Subscript r Baseline asterisk upper G which moves fewer than k vertices, so we have a homomorphism $f: F_r \ast G\to S_p$ f colon upper F Subscript r Baseline asterisk upper G right arrow upper S Subscript p Baseline with $f(F_r \ast G)=S_p$ f left parenthesis upper F Subscript r Baseline asterisk upper G right parenthesis equals upper S Subscript p or $A_p$ upper A Subscript p by Lemma 3.2. Moreover, since for any $j=1, \ldots , m$ j equals 1 comma ellipsis comma m ,

$$ \begin{align*}\gamma_j\notin H=\mathrm{Lab}(\mathrm{Cay}(F_r \ast G,H),H \cdot 1),\end{align*} $$ gamma Subscript j Baseline not an element of upper H equals upper L a b left parenthesis upper C a y left parenthesis upper F Subscript r Baseline asterisk upper G comma upper H right parenthesis comma upper H dot 1 right parenthesis comma

the path $p^{\prime }_j$ p prime Subscript j is not closed in $\Gamma $ normal upper Gamma and, hence, not closed again in $\mathrm {Cay}(F_r \ast G,H_0)$ upper C a y left parenthesis upper F Subscript r Baseline asterisk upper G comma upper H 0 right parenthesis . Therefore, the action of $\gamma _j$ gamma Subscript j on $\mathrm {V}(\mathrm {Cay}(F_r \ast G,H_0,H_0\cdot 1))$ normal upper V left parenthesis upper C a y left parenthesis upper F Subscript r Baseline asterisk upper G comma upper H 0 comma upper H 0 dot 1 right parenthesis right parenthesis does not fix the base point $H_0 \cdot 1$ upper H 0 dot 1 while the action of H does. This implies that $f(\gamma _j) \notin f(H)$ f left parenthesis gamma Subscript j Baseline right parenthesis not an element of f left parenthesis upper H right parenthesis for $1 \leq j \leq m$ 1 less than or equals j less than or equals m .

Lemma 5.3. If $H \cap F_r^{\gamma }\neq 1$ upper H intersection upper F Subscript r Superscript gamma Baseline not equals 1 has infinite index in $F_r^{\gamma }$ upper F Subscript r Superscript gamma for some $\gamma \in F_r \ast G$ gamma element of upper F Subscript r Baseline asterisk upper G , then there is an $F_r$ upper F Subscript r -component in $\Gamma $ normal upper Gamma which is not a cover of $F_r$ upper F Subscript r .

Proof. By Lemma 5.1, $H=\mathrm {Lab}(\Gamma , v_0)$ upper H equals upper L a b left parenthesis normal upper Gamma comma v 0 right parenthesis with base point $v_0:=H \cdot 1$ v 0 colon equals upper H dot 1 . Combining the Kurosh subgroup theorem with Lemma 4.4 shows that there is an X-component C of $\Gamma $ normal upper Gamma such that $\gamma \mathrm {Lab}(C,v)\gamma ^{-1}=H \cap F_r^{\gamma }$ gamma upper L a b left parenthesis upper C comma v right parenthesis gamma Superscript negative 1 Baseline equals upper H intersection upper F Subscript r Superscript gamma , where $\gamma =\mathrm {Lab}(P)$ gamma equals upper L a b left parenthesis upper P right parenthesis for P an approach path from the base point $v_0$ v 0 to the base point v of C. Hence, the index

$$ \begin{align*}[F_r:\mathrm{Lab}(C,v)]=[F_r^\gamma : \gamma\mathrm{Lab}(C,v)\gamma^{-1}]=[F_r^\gamma : H \cap F_r^\gamma]=\infty.\end{align*} $$ StartLayout 1st Row left bracket upper F Subscript r Baseline colon upper L a b left parenthesis upper C comma v right parenthesis right bracket equals left bracket upper F Subscript r Superscript gamma Baseline colon gamma upper L a b left parenthesis upper C comma v right parenthesis gamma Superscript negative 1 Baseline right bracket equals left bracket upper F Subscript r Superscript gamma Baseline colon upper H intersection upper F Subscript r Superscript gamma Baseline right bracket equals infinity period EndLayout

It follows that the X-component C is not a finite cover of $F_r$ upper F Subscript r . In fact, C is not a cover of $F_r$ upper F Subscript r because it is finite.

Proof of Theorem 1.5

By Proposition 5.2, we seek a subgroup $H_0\leq F_r\ast G$ upper H 0 less than or equals upper F Subscript r Baseline asterisk upper G , or equivalently, a cover $\widetilde {\Gamma }:=\mathrm {Cay}(F_r\ast G,H_0,H_0\cdot 1)$ normal upper Gamma overtilde colon equals upper C a y left parenthesis upper F Subscript r Baseline asterisk upper G comma upper H 0 comma upper H 0 dot 1 right parenthesis , whose index $[F_r\ast G: H_0]=|\mathrm {V}(\widetilde \Gamma )|$ left bracket upper F Subscript r Baseline asterisk upper G colon upper H 0 right bracket equals StartAbsoluteValue normal upper V left parenthesis normal upper Gamma overtilde right parenthesis EndAbsoluteValue is a sufficiently large prime p, such that $\Gamma $ normal upper Gamma in (5.1) embeds into the finite cover $\widetilde \Gamma $ normal upper Gamma overtilde and there exists an element $\gamma \in F_r\ast G$ gamma element of upper F Subscript r Baseline asterisk upper G moving fewer than a certain number of vertices in $\widetilde {\Gamma }$ normal upper Gamma overtilde . We construct $\widetilde \Gamma $ normal upper Gamma overtilde in the following four steps.

Step 1. Embedding of each monochromatic component into a finite cover. Since $\Gamma $ normal upper Gamma is finite, there must be finitely many monochromatic components in $\Gamma $ normal upper Gamma ; we denote them by $\Gamma _1,\ldots ,\Gamma _\ell $ normal upper Gamma 1 comma ellipsis comma normal upper Gamma Subscript script l Baseline .

If hypothesis (1) holds, either there is no $F_r$ upper F Subscript r -component in $\Gamma $ normal upper Gamma (that is, $\Gamma $ normal upper Gamma is Y-monochromatic) or every $F_r$ upper F Subscript r -component in $\Gamma $ normal upper Gamma is a finite tree. In the former case, we denote the base point $H\cdot 1$ upper H dot 1 by $\Gamma _1$ normal upper Gamma 1 and re-index the only Y-monochromatic component as $\Gamma _2$ normal upper Gamma 2 . In the latter case, we assume $\Gamma _1$ normal upper Gamma 1 is such an $F_r$ upper F Subscript r -component and not a cover of $F_r$ upper F Subscript r .

If hypothesis (2) holds, then by Lemma 5.3, there is an $F_r$ upper F Subscript r -component in $\Gamma $ normal upper Gamma which is not a cover of $F_r$ upper F Subscript r . Without loss of generality, we again assume it is $\Gamma _1$ normal upper Gamma 1 .

Recall that each finite connected labelled component $\Gamma _i$ normal upper Gamma Subscript i is $F_r$ upper F Subscript r - or G-based, so that $\Gamma _i$ normal upper Gamma Subscript i can be embedded in a cover of $F_r$ upper F Subscript r or G by Lemma 2.3. Moreover, by Theorem 2.4, since both $F_r$ upper F Subscript r and G are LERF, every $\Gamma _i$ normal upper Gamma Subscript i can be embedded in a finite cover $\widetilde \Gamma _i$ normal upper Gamma overtilde Subscript i of $F_r$ upper F Subscript r or G.

Step 2. Construction of $\Gamma ^\ast $ normal upper Gamma Superscript asterisk . We now consider the pushout of the embedding $\alpha _2: \bigcup _{i\geq 2}\Gamma _i\hookrightarrow \bigcup _{i\geq 2}\widetilde \Gamma _i$ alpha 2 colon union Underscript i greater than or equals 2 Endscripts normal upper Gamma Subscript i Baseline right arrow with hook union Underscript i greater than or equals 2 Endscripts normal upper Gamma overtilde Subscript i Baseline :

Commutative square: Top-left union of Gamma_i maps via alpha_1 to Gamma (top-right) and via alpha_2 to union of Gamma-tilde_i (bottom-left). Both right and bottom terms map to Gamma* (bottom-right).

Note that every $\Gamma _i$ normal upper Gamma Subscript i embeds in $\widetilde \Gamma _i$ normal upper Gamma overtilde Subscript i and $\Gamma _i$ normal upper Gamma Subscript i is a monochromatic component of $\Gamma $ normal upper Gamma , so no foldings occur between distinct $\widetilde \Gamma _i$ normal upper Gamma overtilde Subscript i . It follows that $\Gamma _i$ normal upper Gamma Subscript i embeds in $\Gamma ^\ast $ normal upper Gamma Superscript asterisk . Moreover, every monochromatic component of $\Gamma ^\ast $ normal upper Gamma Superscript asterisk is $F_r$ upper F Subscript r - or G-based. Hence, $\Gamma ^\ast $ normal upper Gamma Superscript asterisk is $F_r\ast G$ upper F Subscript r Baseline asterisk upper G -based by Lemma 2.6 (see Figure 2).

Figure 2

An example for $\Gamma $ normal upper Gamma and $\Gamma ^\ast $ normal upper Gamma Superscript asterisk when $X=\{x_1,x_2\}$ upper X equals StartSet x 1 comma x 2 EndSet , $Y=\{ y_1,y_2 \}$ upper Y equals StartSet y 1 comma y 2 EndSet . We take $H=\langle y_1x_1^{-1}y_1, x_1x_2x_1^{-1}\rangle $ upper H equals left angle bracket y 1 x 1 Superscript negative 1 Baseline y 1 comma x 1 x 2 x 1 Superscript negative 1 Baseline right angle bracket and $\gamma _1=y_2$ gamma 1 equals y 2 .

Two directed graph panels compare edge structures and node connections for x one, x two, y one, y two, H, H x one, H y one, H y two, and H y one inverse. See long description.
Figure 2 Long description

The left panel shows a directed graph with central node H. Outward, three black nodes are connected by black arrows labeled x one and x two, and blue arrows labeled y one and y two. H connects to H x one by x one, to H y one by y one, and to H y two by y two. H y one connects to H y one inverse by x one, and H x one has a loop labeled x two. H y two is at the far left, connected by y two. The right panel expands the structure: H remains central, with additional loops and connections. H connects to H x one by x one, to H y one by y one, and to H y two by y two. H y one connects to H y one inverse by x one, and H x one has a loop labeled x two. H y two has a loop labeled y two. Multiple blue arrows labeled y one and y two form cycles among H, H y one, H y one inverse, and H y two. Black arrows labeled x one and x two form cycles among H, H x one, and H y one inverse. All labels are typeset in italic.

Step 3. Modify $\Gamma _1$ normal upper Gamma 1 to obtain a precover $\Gamma '$ normal upper Gamma prime of $F_r\ast G$ upper F Subscript r Baseline asterisk upper G . Denote $|\mathrm {V}(\Gamma ^\ast )|=k$ StartAbsoluteValue normal upper V left parenthesis normal upper Gamma Superscript asterisk Baseline right parenthesis EndAbsoluteValue equals k . Since the labelled graph $\Gamma _1$ normal upper Gamma 1 is $F_r$ upper F Subscript r -based and not a cover, it is not an $X^\ast $ upper X Superscript asterisk -saturated graph by Lemma 2.3. From Wilton’s construction in Lemma 3.3, after attaching some graphs $W_n$ upper W Subscript n and $V_s$ upper V Subscript s , the graph $\Gamma _1$ normal upper Gamma 1 can embed in a cover $\widetilde \Gamma _1$ normal upper Gamma overtilde Subscript 1 of $F_r$ upper F Subscript r with

$$ \begin{align*}|\mathrm{V}(\widetilde\Gamma_1)|=|\mathrm{V}(\Gamma_1)|+n+4.\end{align*} $$ StartLayout 1st Row StartAbsoluteValue normal upper V left parenthesis normal upper Gamma overtilde Subscript 1 Baseline right parenthesis EndAbsoluteValue equals StartAbsoluteValue normal upper V left parenthesis normal upper Gamma 1 right parenthesis EndAbsoluteValue plus n plus 4 period EndLayout

We now take a sufficiently large prime p for $k+5$ k plus 5 as in Lemma 3.2 and let $n=p-k-4$ n equals p minus k minus 4 . We construct the pushout of the embedding $\alpha _2':\Gamma _1 \hookrightarrow \widetilde \Gamma _1$ alpha prime 2 colon normal upper Gamma 1 right arrow with hook normal upper Gamma overtilde Subscript 1 Baseline as shown:

Commutative diagram: Γ₁ maps via α’₁ to Γ* and α’₂ to Γ̃₁. Arrows from Γ* and Γ̃₁ both point to Γ’.

Since $\Gamma _1$ normal upper Gamma 1 embeds in $\widetilde \Gamma _1$ normal upper Gamma overtilde Subscript 1 and $\widetilde \Gamma _1$ normal upper Gamma overtilde Subscript 1 is monochromatic, no foldings occur between $\widetilde \Gamma _1$ normal upper Gamma overtilde Subscript 1 and $\Gamma ^\ast $ normal upper Gamma Superscript asterisk , and both $\Gamma ^\ast $ normal upper Gamma Superscript asterisk and $\widetilde \Gamma _1$ normal upper Gamma overtilde Subscript 1 embed in $\Gamma '$ normal upper Gamma prime . Since $\Gamma ^\ast $ normal upper Gamma Superscript asterisk is $F_r \ast G$ upper F Subscript r Baseline asterisk upper G -based and $\widetilde \Gamma _1$ normal upper Gamma overtilde Subscript 1 is $F_r$ upper F Subscript r -based, $\Gamma '$ normal upper Gamma prime is $F_r\ast G$ upper F Subscript r Baseline asterisk upper G -based by Lemma 2.6 and every monochromatic component is a cover. Thus, $\Gamma '$ normal upper Gamma prime is a precover of $F_r\ast G$ upper F Subscript r Baseline asterisk upper G .

Step 4. Construction of the final cover $\widetilde {\Gamma }$ normal upper Gamma overtilde . By Lemma 2.8, the precover $\Gamma '$ normal upper Gamma prime can embed in a cover $\widetilde {\Gamma }$ normal upper Gamma overtilde of $F_r \ast G$ upper F Subscript r Baseline asterisk upper G without adding new vertices, so $\Gamma $ normal upper Gamma itself embeds in the cover $\widetilde {\Gamma }$ normal upper Gamma overtilde with $|\mathrm {V}(\widetilde {\Gamma })|=p=k+n+4$ StartAbsoluteValue normal upper V left parenthesis normal upper Gamma overtilde right parenthesis EndAbsoluteValue equals p equals k plus n plus 4 . Since $x_2$ x 2 acts trivially on $\mathrm {V}(W_n)$ normal upper V left parenthesis upper W Subscript n Baseline right parenthesis and acts nontrivially on $\mathrm {V}(V_s)$ normal upper V left parenthesis normal upper V Subscript s Baseline right parenthesis , there must be fewer than $k+5$ k plus 5 vertices (lying in $\Gamma ^\ast \cup V_s$ normal upper Gamma Superscript asterisk Baseline union upper V Subscript s ) in $\widetilde {\Gamma }$ normal upper Gamma overtilde which are moved by $x_2$ x 2 . Therefore, the cover $\widetilde {\Gamma }$ normal upper Gamma overtilde is precisely what we need and the proof is complete.

Acknowledgements

The authors would like to thank Qian Chen very much for his comments. Thanks are also extended to the anonymous referee.

Footnotes

The authors are partially supported by NSFC (No. 12471066) and the Shaanxi Fundamental Science Research Project for Mathematics and Physics (No.23JSY027).

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Figure 0

Figure 1 The graph W4$W_4$ on the left and V+1,−1$V_{+1,-1}$ on the right.Figure 1 long description.

Figure 1

Figure 2 An example for Γ$\Gamma $ and Γ∗$\Gamma ^\ast $ when X={x1,x2}$X=\{x_1,x_2\}$, Y={y1,y2}$Y=\{ y_1,y_2 \}$. We take H=⟨y1x1−1y1,x1x2x1−1⟩$H=\langle y_1x_1^{-1}y_1, x_1x_2x_1^{-1}\rangle $ and γ1=y2$\gamma _1=y_2$.Figure 2 long description.