2. Preliminaries

We start this chapter by reviewing some standard facts of the word length and length functions.

We note that for a given group G and a finite generating set S consisting of elements of order two we have $l(g)=l_{sy}(g)$ for all $g\in G$.

The length of $g\in G$ is closely connected to the length of a special path in a geometric object which is associated to the group G, the Cayley-graph $Cay(G,S)$. Before we give a definition of this graph we recall the definition and some important facts about general graphs which we will need later on.

A graph Γ is a pair $(V(\Gamma),E(\Gamma))$ where $V(\Gamma)$ is a set whose elements are called vertices and $E(\Gamma)$ is a subset of $\mathcal{P}_2(V):=\left\{X\mid X\subseteq V, |X|=2\right\}$ whose elements are called edges. Usually, graphs are visualized graphically, where we draw for each vertex $x\in V(\Gamma)$ a point and label it with x and two points $x, y$ are connected by a line if $\left\{x, y\right\}\in E(\Gamma)$.

Given a graph Γ and a vertex $x\in V(\Gamma)$ we define two subsets of $V(\Gamma)$ that are associated to x. The link of x, denoted by lk(x) is defined as $lk(x):=\left\{y\in V(\Gamma)\mid \left\{x,y\right\}\in E(\Gamma)\right\}$ and the star of x, denoted by st(x) is defined as $st(x):=lk(x)\cup\left\{x\right\}$. A graph Γ is called complete if $st(x)=V(\Gamma)$ for all $x\in V(\Gamma)$. A subgraph $\Omega\subseteq\Gamma$ is called full if for all pair of vertices $(v,w)\in V(\Omega)\times V(\Omega)$ we have $\left\{v,w\right\}\in E(\Omega)$ if and only if $\left\{v,w\right\}\in E(\Gamma)$.

Let G be a group and let S be a generating set for G. The Cayley-graph for G with respect to S, denoted by $Cay(G,S)$ is a graph with vertex set $V(Cay(G,S))=G$ and edge set $E(Cay(G,S))=\left\{\left\{g, gs\right\}\mid g\in G, s\in S\cup S^{-1}\right\}$. The distance between two vertices is defined as a number of edges in a shortest path connecting those vertices. Note that l(g) is equal to the distance between the vertices 1_G and g. Hence the number of elements in G with word length n is equal to the number of vertices in the sphere with centre 1_G of radius n in $Cay(G,S)$. Let us consider the Cayley-graph of the free group F ₂ with the generating set $\left\{x,y\right\}$ in Figure 1.

Figure 1.

$Cay(F_2,\left\{x,y\right\}).$

For example, the number of elements in F ₂ with length 2 is equal to 12. One geometric way to count the elements in F ₂ is by counting the vertices in the sphere with centre 1 of radius n in the Cayley-graph $Cay(F_2,\left\{x,y\right\})$. Let a_n be this number. We get the sequence $(a_n)_{n\in\mathbb{N}}$ where $a_0=1$ and $a_n=4\cdot 3^{n-1}$ for $n\geq 1$. We convert this sequence into a formal power series $1+a_1t+a_2t^2+\ldots $ which leads us to the definition of the spherical growth series. The best general reference on growth series of groups is [Reference Mann11].

We note that $\mathcal{G}_{(G,S)}(t)$ is an element in $\mathbb{Z}[[t]]$ the ring of formal power series in the variable t over $\mathbb{Z}$. We now give some examples.

Given two groups G and H with finite generating sets S_G resp. S_H, to construct a new group using given ones it is natural to use direct or free product construction. For direct and free products, there are formulas for the spherical growth series in terms of spherical growth series of the factors [Reference Mann11].

\begin{gather*} \mathcal{G}_{(G\times H,S_G\cup S_H)}(t)=\mathcal{G}_{(G,S_G)}(t)\cdot \mathcal{G}_{(H,S_H)}(t)\,,\\ \frac{1}{\mathcal{G}_{(G*H, S_G\cup S_H)}(t)}=\frac{1}{\mathcal{G}_{(G, S_G)}(t)}+ \frac{1}{\mathcal{G}_{(H, S_H)}(t)}-1. \end{gather*}

A generalization of direct resp. free product construction is amalgamated products and graph products of groups. Let us recall a formula for the spherical growth series of an amalgamated product. First, we need a definition.

It was proven in [Reference Lewin10] that if (L, R) is admissible in (H, S) and in (K, T), then the spherical growth series of $G=H*_L K$ can be computed using smaller pieces of G.

Proposition 2.5. If (L, R) is admissible in (H, S) and in (K, T), then

\begin{equation*}\frac{1}{\mathcal{G}_{(H*_L K, S\cup T)}(t)}=\frac{1}{\mathcal{G}_{(H, S)}(t)}+\frac{1}{\mathcal{G}_{(K, T)}(t)}-\frac{1}{\mathcal{G}_{(L, R)}(t)}.\end{equation*}

Now we move on to graph products of groups. Given a finite graph Γ and a collection of finitely generated groups G_x for $x\in V(\Gamma)$, the graph product of groups is defined as:

\begin{equation*}G_\Gamma=\big(\ast_{x\in V(\Gamma)} G_x\big)/\langle\langle [g,h]\mid g\in G_x, h\in G_y, \left\{x,y\right\}\in E(\Gamma)\rangle\rangle.\end{equation*}

We note that, if Γ is discrete, then the associated graph product of groups is the free product of the vertex groups and if Γ is complete, then the associated graph product of groups is the direct product of the vertex groups. If all vertex groups are infinite cyclic, then we call $G_\Gamma$ a right-angled Artin group. For every vertex group G_x let S_x be a finite generating set and we set $S:=\bigcup_{x\in V(\Gamma)} S_x$. A formula for the spherical growth series of $G_\Gamma$ in terms of the spherical growth series of the vertex groups was proven in [Reference Lewin10] for isomorphic vertex groups. Here we recall a special case of this formula where $G_\Gamma$ is a right-angled Artin group and $S_x=\left\{1\right\}$ for every $x\in V$. Let c_i be the number of complete subgraphs in Γ on i vertices. Then

\begin{equation*}\frac{1}{\mathcal{G}_{(G_\Gamma,S)}(t)}=\sum_i (-1)^i c_i \frac{(\frac{1+t}{1-t}-1)^i}{(\frac{1+t}{1-t})^i}.\end{equation*}

This formula was generalized for arbitrary vertex groups in [Reference Chiswell6]. Let $G_\Gamma$ be a graph product of finitely generated vertex groups. We define for each complete subgraph $\Delta\subseteq\Gamma$, $P_\Delta(t):=\prod_{x\in V(\Delta)}(\frac{1}{\mathcal{G}_{(G_x, S_x)}(t)}-1)$. Then

\begin{equation*}\frac{1}{\mathcal{G}_{(G_\Gamma, S)}(t)}=\sum P_\Delta(t),\end{equation*}

where the summation is taken over all complete subgraphs of Γ including the empty one for which $P_\emptyset(t)=1$.

We want also to point out that the geodesic growth series of graph products of cyclic groups was studied in [Reference Marjanski, Solon, Zheng and Zopff12].

Further groups for which it is possible to compute the spherical growth series using smaller building blocks of the group are Coxeter groups. Coxeter groups have special subgroups which can be considered as building blocks for the whole group. Given a finite graph Γ with an edge-labelling $m\colon E(\Gamma)\to\mathbb{N}_{\geq 2}$. The Coxeter group associated to Γ is given by the presentation:

\begin{equation*}W=\langle V(\Gamma)\mid x^2\,\text{for all }x\in V(\Gamma), (xy)^{m(\left\{x,y\right\})}\,\text{for all } \left\{x,y\right\}\in E(\Gamma)\rangle.\end{equation*}

For any subset $X\subseteq V(\Gamma)$ the subgroup generated by the set X is canonically isomorphic to the Coxeter group which is associated to the full subgraph of Γ with the vertex set X. This subgroup is called a standard parabolic subgroup and we denote it by W_X. A natural question is if it is possible to use the spherical growth series of special parabolic subgroups to obtain a formula for the spherical growth series of the whole group. It was proven in [Reference Solomon17], [Reference Bourbaki2] that it is indeed the case. Let (W, S) be a Coxeter system. If W is finite, then

\begin{equation*}\frac{t^m+(-1)^{|S|+1}}{\mathcal{G}_{(W,S)}(t)}=\sum_{X\subsetneq S}\frac{(-1)^{|X|}}{\mathcal{G}_{(W_X,X)}(t)},\end{equation*}

where $m=max\left\{l(w)\mid w\in W\right\}$. The spherical growth series of a finite Coxeter group can also be calculated using the non-recursive formula:

\begin{equation*}\mathcal{G}_{(W,S)}(t)=\prod_{i=1}^k(1+t+\ldots+t^{m_i}),\end{equation*}

where $m_1,\dots,m_k$ are the exponents of (W, S).

If W is infinite, then

\begin{equation*}\frac{(-1)^{|S|+1}}{\mathcal{G}_{(W,S)}(t)}=\sum_{X\subsetneq S}\frac{(-1)^{|X|}}{\mathcal{G}_{(W_X,X)}(t)}.\end{equation*}

In particular, the above formulas show that for a Coxeter group W there exists a polynomial f(t) such that:

\begin{equation*}\frac{f(t)}{\mathcal{G}_{(W,S)}(t)}=\sum_{X\subsetneq S}\frac{(-1)^{|X|}}{\mathcal{G}_{(W_X,X)}(t)}.\end{equation*}

3. Dyer groups

We begin this chapter with the definition of the main protagonist in this article, a Dyer group. For two letters $a, b$ and a natural number m we define $\pi(a,b,m):= abababa\ldots$ where the length of the word is m. For example $\pi(a,b,3)=aba$.

3.1. Dyer tools

We start by recalling several results which were proven by Dyer in [Reference Dyer7]. Let G be a group and $g\in G$. We denote the order of g by o(g). If o(g) is finite, then we write $\mathbb{Z}_{o(g)}$ for the cyclic group of cardinality o(g) and if o(g) is infinite, then we write $\mathbb{Z}_{o(g)}$ for the infinite cyclic group. More generally we use the notation $\mathbb{Z}_n=\mathbb{Z}/n\mathbb{Z}$ if n is a positive integer and $\mathbb{Z}_\infty=\mathbb{Z}$.

Let (D, V) be a Dyer system. By definition, a conjugate of a generator $x\in V$ is called a reflection. We define

\begin{equation*}R:=\left\{gxg^{-1}\mid g\in D, x\in V\right\}.\end{equation*}

R is the set of all reflections in D. For $\rho\in R$ we define a copy of $\mathbb{Z}_{o(\rho)}$ as $H_\rho=\left\{a[\rho]\mid a\in\mathbb{Z}_{o(\rho)}\right\}$. The set H_ρ is an abelian group whose group operation is defined by $a[\rho]+b[\rho]:=(a+b)[\rho]$. Hence H_ρ is isomorphic to $\mathbb{Z}_{o(\rho)}$. Further, we define

\begin{equation*}M=\bigoplus\limits_{\rho\in R}H_\rho.\end{equation*}

This set is an abelian group with canonical group operation $\sum a_\rho[\rho]+\sum b_\rho[\rho]=\sum (a_\rho+b_\rho)[\rho]$. Furthermore, this abelian group is a D-module where the structure of the D-module is defined for $g\in D$ by:

\begin{equation*}g\cdot \sum a_\rho[\rho]:=\sum a_\rho[g\rho g^{-1}].\end{equation*}

Let $g\in D$. We pick one syllabic representative $(x_1^{a_1}, x_2^{a_2}, \ldots, x_l^{a_l})$ for g, that is, a tuple of syllables such that $g=x_1^{a_1}x_2^{a_2}\ldots x_l^{a_l}$. For each $i\in\left\{1,\ldots, l\right\}$ we define a reflection:

\begin{equation*}\rho_i:=x_1^{a_1}x_2^{a_2}\ldots x^{a_{i-1}}_{i-1}x_ix_{i-1}^{-a_{i-1}}\ldots x_2^{-a_2}x_1^{-a_1}.\end{equation*}

We set

\begin{equation*}N(g)=\sum_{i=1}^l a_i[\rho_i]\in M.\end{equation*}

For $n\in\mathbb{N}_{\ge 2}\cup\{\infty\}$ and $a\in\mathbb{Z}_n$ we denote by $\lVert a\rVert_n$ the word length of a with respect to the generating set $\left\{1\right\}$.

Theorem 3.2. ([Reference Dyer7]) Let (D, V) be a Dyer system. Let $g,h\in D$.

1. (1) N(g) does not depend on the choice of the syllabic representative for g.

2. (2) Let $N(g)=\sum_{\rho\in R}a_\rho(g)[\rho]$. Then

   1. (a) $l_{sy}(g)=\mid\left\{\rho\in R\mid a_\rho(g)\neq 0\right\}\mid$.

   2. (b) $l(g)=\sum_{\rho\in R}\lVert a_\rho(g)\rVert_{o(\rho)}$.

3. (3) $N(gh)=N(g)+g\cdot N(h)$.

Let $g\in D$. A syllabic representative $(x_1^{a_1},x_2^{a_2},\dots,x_l^{a_l})$ for g is called reduced if $l=l_{sy}(g)$. The following is a direct consequence of part (2) of Theorem 3.2 and it will be often used hereafter.

Corollary 3.3. Let $g\in D$ and $(x_1^{a_1},x_2^{a_2},\dots,x_l^{a_l})$ be a reduced syllabic representative for g. Then

\begin{equation*}l(g)=\|a_1\|_{o(x_1)}+\|a_2\|_{o(x_2)}+\ldots+\|a_l\|_{o(x_l)}.\end{equation*}

Proof. For each $i\in\left\{1,\ldots,l\right\}$ we set

\begin{equation*}\rho_i=x_1^{a_1}x_2^{a_2}\ldots x_{i-1}^{a_{i-1}}x_ix_{i-1}^{-a_{i-1}}\ldots x_2^{-a_2}x_1^{-a_1}.\end{equation*}

Then

\begin{equation*}N(g)=\sum_{i=1}^la_i[\rho_i].\end{equation*}

Since $l=l_{sy}(g)$, by Theorem 3.2 (2–a) we have $\rho_i\neq\rho_j$ for i ≠ j. By Theorem 3.2 (2–b) it follows that:

\begin{equation*}l(g)=\|a_1\|_{o(\rho_1)}+\|a_2\|_{o(\rho_2)}+\ldots+\|a_l\|_{o(\rho_l)} =\|a_1\|_{o(x_1)}+\|a_2\|_{o(x_2)}+\ldots+\|a_l\|_{o(x_l)}.\end{equation*}

3.2. Standard parabolic subgroups

Let (D, V) be a Dyer system. For any subset $X\subseteq V$, we denote the subgroup generated by the set X by $D_X\subseteq D$. D_X is called the standard parabolic subgroup generated by X.

Let $(\Gamma,m,f)$ be the Dyer graph associated with (D, V). We denote by $\Gamma_X$ the full subgraph of Γ spanned by X, by m_X the restriction of m to $E(\Gamma_X)$, and by f_X the restriction of f to $V(\Gamma_X)=X$. Then $(\Gamma_X,m_X,f_X)$ is a Dyer graph and we know by [Reference Dyer7] that $(D_X,X)$ is the Dyer system associated with $(\Gamma_X,m_X,f_X)$ (see also [Reference Paris and Soergel13, Proposition 2.7]).

Lemma 3.4. Let (D, V) be a Dyer system. Let D_X be a standard parabolic subgroup of D. Then for any $g\in D_X$ we have $l_X(g)=l_V(g)$.

Proof. Let $g\in D_X$. Let $(x_1^{a_1},x_2^{a_2},\ldots,x_l^{a_l})$ be a reduced syllabic representative for g. We know by [Reference Paris and Soergel13, Lemma 2.5] that $x_1,x_2,\dots, x_l\in X$, hence, by Corollary 3.3,

\begin{equation*}l_V(g)=\|a_1\|_{o(x_1)}+\|a_2\|_{o(x_2)}+\ldots+\|a_l\|_{o(x_l)}\ge l_X (g).\end{equation*}

It is clear that we also have $l_X(g)\ge l_V(g)$, thus $l_X(g)=l_V(g)$.

Proposition 3.5. Let (D, V) be a Dyer system and D_X be a standard parabolic subgroup. Then for every $g\in D$

1. (1) there exist a unique $g_0\in gD_X$ of minimal syllabic length in gD_X and g ₀ satisfies:

   \begin{equation*}l_{sy}(g_0h)=l_{sy}(g_0)+l_{sy}(h)\text{and } l(g_0h)=l(g_0)+l(h)\end{equation*}

   for all $h\in D_X$.

2. (2) there exists a unique $g'_0\in D_Xg$ of minimal syllabic length in D_Xg and g ₀ satisfies

   \begin{equation*}l_{sy}(hg'_0)=l_{sy}(h)+l_{sy}(g'_0)\text{and } l(hg'_0)=l(h)+l(g'_0)\end{equation*}

   for all $h\in D_X$.

Proof. The statements regarding syllabic length were proved in [Reference Paris and Soergel13, Proposition 2.8]. Hence we know that there exists a unique $g_0\in gD_X$ such that for all $h\in D_X$ we have:

\begin{equation*}l_{sy}(g_0h)=l_{sy}(g_0)+l_{sy}(h).\end{equation*}

Let $h\in D_X$. Let $(x_1^{a_1},\ldots,x_p^{a_p})$ be a reduced syllabic representative for g ₀ and $(y_1^{b_1},\ldots,y_q^{b_q})$ be a reduced syllabic representative for h. We know from the above that $(x_1^{a_1},\ldots,x_p^{a_p},y_1^{b_1},\ldots,y_q^{b_q})$ is a reduced syllabic representative for $g_0h$, hence, by Corollary 3.3,

\begin{equation*}l(g_0h)=\|a_1\|_{o(x_1)}+\ldots+\|a_p\|_{o(x_p)}+\|b_1\|_{o(y_1)}+\ldots+\|b_q\|_{o(y_q)}=l(g_0)+l(h).\end{equation*}

The proof of part (2) is the same as for part (1).

Corollary 3.6. Let $(\Gamma,m,f)$ be a Dyer graph and (D, V) be the associated Dyer system. Every pair $(D_X,X)$ where D_X is a standard parabolic subgroup is admissible.

Proof. For a standard parabolic subgroup D_X and an element $g\in D$ there exists a unique $g_0\in gD_X$ such that $l(g_0h)=l(g_0)+l(h)$ for all $h\in D_X$. We take these minimal elements as a transversal. Lemma 3.4 and Proposition 3.5 show that this transversal is admissible.

Corollary 3.7. Let $(\Gamma,m,f)$ be a Dyer graph. Let $v\in V(\Gamma)$. If $st(v)\neq V(\Gamma)$, then

\begin{equation*}D=D_{V-\left\{v\right\}}*_{D_{lk(v)}}D_{st(v)},\end{equation*}

and

\begin{equation*}\frac{1}{\mathcal{G}_{(D, V)}(t)}=\frac{1}{\mathcal{G}_{(D_{V-\left\{v\right\}}, V-\left\{v\right\})}(t)}+\frac{1}{\mathcal{G}_{(D_{st(v)}, st(v))}(t)}-\frac{1}{\mathcal{G}_{(D_{lk(v)}, lk(v))}(t)}.\end{equation*}

Proof. The proof of the equality $D=D_{V-\left\{v\right\}}*_{D_{lk(v)}}D_{st(v)}$ follows by analysing the presentation of D and the canonical presentation of the amalgam. By Corollary 3.6 the groups in this amalgamated product are admissible. Thus by Proposition 2.5 we get the above equality of the spherical growth series.

4. X-minimality

Note that, by Proposition 3.5, if g is X-minimal, then $l_{sy}(gh)=l_{sy}(g)+l_{sy}(h)$ and $l(gh)=l(g)+l(h)$ for all $h\in D_X$. Note also that, if $X\subseteq Y\subseteq V$ and g is Y-minimal, then g is also X-minimal, since $gD_X\subseteq gD_Y$.

Definition 4.2. Let (D, V) be a Dyer system and $X\subseteq V$. We define two subsets of D as follows:

\begin{equation*}A_X=A_X(D):=\left\{g\in D\mid g\text{is }X\text{-minimal}\right\} \text{and } B_X=B_X(D):=A_X-(\cup_{X\subsetneq Y}A_Y).\end{equation*}

An important feature of these sets is that, for $X\subseteq V$, the set A_X is the disjoint union of those B_Y with $X\subseteq Y$. This property is less obvious than it seems and follows from the following lemma.

The following lemma is a particular case of the well-known general Möbius inversion formula (see [Reference Stanley18, Section 3.7] or [Reference Rota14] for example).

Lemma 4.5. Let V be a set and $\mathcal{P}(V)$ be the set of all subsets of V. Further, let G be an abelian group. If the functions $f\colon\mathcal{P}(V)\to G$ and $g\colon\mathcal{P}(V)\to G$ satisfy

\begin{equation*}f(X)=\sum_{X\subseteq Y}g(Y)\text{for all }X\in\mathcal{P}(V),\end{equation*}

then they satisfy

\begin{equation*}g(X)=\sum_{X\subseteq Y}(-1)^{|Y-X|}f(Y)\text{for all }X\in\mathcal{P}(V).\end{equation*}

Proposition 4.6. Let (D, V) be a Dyer system. For $X\subseteq V$ we have

\begin{equation*}\mathcal{G}_{(B_X, V)}(t)=\sum_{X\subseteq Y} (-1)^{|Y-X|}\frac{\mathcal{G}_{(D,V)}(t)}{\mathcal{G}_{(D_Y,Y)}(t)}.\end{equation*}

In particular, for $X=\emptyset$ we obtain:

\begin{equation*}\mathcal{G}_{(B_\emptyset, V)}(t)=\sum_{Y} (-1)^{|Y|}\frac{\mathcal{G}_{(D,V)}(t)}{\mathcal{G}_{(D_Y,Y)}(t)},\end{equation*}

which is equivalent to:

\begin{equation*}\frac{\mathcal{G}_{(B_\emptyset, V)}(t)+(-1)^{|V|+1}}{\mathcal{G}_{(D,V)}(t)}=\sum_{Y\subsetneq V}\frac{(-1)^{|Y|}}{\mathcal{G}_{(D_Y,Y)}(t)}.\end{equation*}

Proof. Let (D, V) be a Dyer system. We define two functions $f\colon\mathcal{P}(V)\to\mathbb{Z}[[t]]$ and $g\colon\mathcal{P}(V)\to\mathbb{Z}[[t]]$ where $\mathbb{Z}[[t]]$ is the formal power series ring with coefficients in the group $\mathbb{Z}$ as follows:

\begin{equation*}f(X)(t)=\sum_{g\in A_X}t^{l(g)}\text{and } g(X)(t)=\sum_{g\in B_X}t^{l(g)}.\end{equation*}

By Corollary 4.4 the set A_X is a disjoint union of those B_Y, where $X\subseteq Y$. Hence we have

\begin{equation*}f(X)(t)=\sum_{X\subseteq Y}g(Y)(t),\end{equation*}

and by Lemma 4.5 we obtain

\begin{equation*}g(X)(t)=\sum_{X\subseteq Y}(-1)^{|Y-X|}f(Y)(t).\end{equation*}

By definition we have $g(X)(t)=\mathcal{G}_{(B_X,V)}(t)$. Thus we obtain

\begin{equation*}\mathcal{G}_{(B_X,V)}(t)=\sum_{X\subseteq Y}(-1)^{|Y-X|}f(Y)(t).\end{equation*}

Further, $D=\cup_{g\in A_Y}gD_Y$ and this union is disjoint. More precisely, two cosets are equal or disjoint. Assume that there exist $g_1, g_2\in A_Y$, $g_1\neq g_2$ such that $g_1D_Y=g_2D_Y$. Since g ₁ and g ₂ are both Y-minimal it follows by Proposition 3.5 that $g_1=g_2$, contradiction.

We obtain

\begin{equation*}\mathcal{G}_{(D,V)}(t)=\sum_{g\in A_Y}\sum_{u\in D_Y}t^{l(g)+l(u)}=f(Y)(t)\cdot \mathcal{G}_{(D_Y,V)}(t).\end{equation*}

Finally, by Lemma 3.4 we have $\mathcal{G}_{(D_Y,V)}(t)=\mathcal{G}_{(D_Y,Y)}(t)$, hence

\begin{equation*}\mathcal{G}_{(B_X,V)}(t)=\sum_{X\subseteq Y}(-1)^{|Y-X|}f(Y)(t)=\sum_{X\subseteq Y}(-1)^{|Y-X|}\frac{\mathcal{G}_{(D,V)}(t)}{\mathcal{G}_{(D_Y,Y)}(t)}.\end{equation*}

Our next task is to give a good description of the set $B_\emptyset$. We are particularly interested in properties of (D, V) that ensure the set $B_\emptyset$ to be empty.

Note that, if D is of spherical type, then $D=D_2\times D_p\times D_\infty$, $D_p=\prod_{x\in V_p} \mathbb{Z}/f(x)\mathbb{Z}$, and $D_\infty=\mathbb{Z}^l$ where $l=|V_\infty|$.

The description of $B_\emptyset$ when $D=D_2$ is a Coxeter group is well-known and it is a direct consequence of the following.

The element w ₀ of Proposition 4.8 is called the longest element of W, if it exists. The following is a straightforward consequence of Proposition 4.8.

In the general case we have the following.

Proof. We first prove that, if $B_\emptyset\neq\emptyset$, then D is of spherical type. We will then show that, if D is of spherical type, then $B_\emptyset\neq\emptyset$ and the elements of $B_\emptyset$ are as described in part (2).

Suppose $B_\emptyset\neq\emptyset$. This means that there exists $g\in D$ such that $g\not\in A_{\{x\}}$ for all $x\in V$. So, we can pick $g\in D$ such that, for all $x\in V$, there exists $a\in\mathbb{Z}_{f(x)}-\{0\}$ such that $l_{sy}(gx^a)\le l_{sy}(g)$. We start by showing that Γ is complete. Let $x,y\in V$, x ≠ y. Set $X=\{x,y\}$. We know that there exist $a\in\mathbb{Z}_{f(x)}-\{0\}$ and $b\in\mathbb{Z}_{f(y)}-\{0 \}$ such as $l_{sy}(gx^a)\le l_{sy}(g)$ and $l_{sy}(gy^b)\le l_{sy}(g)$. On the other hand by Proposition 3.5, there exist $g_0\in A_X$ and $h\in D_X$ such that $g=g_0h$. By Proposition 3.5 we have

\begin{equation*}l_{sy}(g)=l_{sy}(g_0)+l_{sy}(h)\,,\ l_{sy}(gx^a)=l_{sy}(g_0)+l_{sy }(hx^a)\,,\ l_{sy}(gy^b)=l_{sy}(g_0)+ l_{sy}(hy^b).\end{equation*}

Thus, $l_{sy}(hx^a)\le l_{sy}(h)$ and $l_{sy}(hy^b)\le l_{sy}(h)$. If x and y are not connected by an edge, then $D_X=\mathbb{Z}_{f(x)}*\mathbb{Z}_{f(y)}$ and there is no h in $\mathbb{Z}_{f(x)}*\mathbb{Z}_{f(y)}$ such that $l_{sy}(hx^a)\le l_{sy}(h)$ and $l_{sy}(hy^b)\le l_{sy}(h)$. So, x and y are connected by an edge.

Since Γ is a complete graph, we have $D=D_2\times D_p\times D_\infty$, $D_p=\prod_{x\in V_p}\mathbb{Z}/f(x)\mathbb{Z} $, and $D_\infty=\mathbb{Z}^l$, where $l=|V_\infty|$. Assume that $B_\emptyset\neq\emptyset$. Let $g\in B_\emptyset$. We write $g=g_2g_pg_\infty$ with $g_2\in D_2$, $g_p\in D_p$, and $g_\infty\in D_\infty$. For each $x\in V_2$ we have

\begin{equation*}l(g_2x)+l_{sy}(g_p)+l_{sy}(g_\infty)=l_{sy}(gx) \lt l_{sy}(g)=l(g_2)+l_{sy}(g_p)+l_{sy}(g_\infty),\end{equation*}

hence $l(g_2x) \lt l(g_2)$.

By Proposition 4.8 this implies that D ₂ is a finite Coxeter group and g ₂ is the longest element of D ₂. So, if $B_\emptyset\neq\emptyset$, then D is of spherical type.

Suppose now that D is of spherical type. Then $D=D_2\times D_p\times D_\infty$, D ₂ is a finite Coxeter group, $D_p=\prod_{x\in V_p}\mathbb{Z}/f(x)\mathbb{Z }$, and $D_\infty=\mathbb{Z}^l$, where $l=|V_\infty|$. Set $V_p=\{x_1,\dots,x_k\}$ and $V_\infty=\{y_1,\dots,y_l\}$. Let $g\in B_\emptyset$. Write g in the form $g=wx_1^{a_1}\ldots x_k^{a_k}y_1^{b_1}\ldots y_l^{b_l}$ with $w\in D_2$, $a_i\in\mathbb{Z}/f(x_i)\mathbb{Z} $ for all $i\in\{1,\dots,k\}$, and $b_j\in\mathbb{Z}$ for all $j\in\{1,\ldots,l\}$. Let $i\in\{1,\ldots,k\}$. If we had $a_i=0$, then we would have $l_{sy}(gx_i^c) \gt l_{sy}(g)$ for all $c\in(\mathbb{Z}/f(x_i)\mathbb {Z})-\{0\}$, hence we would have $g\not\in B_\emptyset$. So $a_i\neq0$ for all $i\in\{1,\dots,k\}$. Similarly, $b_j\neq0$ for all $j\in\{1,\ldots,l\}$. If w were not the longest element of D ₂, then there would exist $x\in V_2$ such that $l(wx) \gt l(w)$, hence there would exist $x\in V_2$ such that $l_{sy}(gx) \gt l_{sy}(g)$. So, w is the longest element of D ₂.

Let $g\in D$ which can be written in the form $g=w_0x_1^{a_1}\ldots x_k^{a_k}y_1^{b_1}\ldots y_l^{b_l}$, where w ₀ is the longest element of D ₂, $a_i\in(\mathbb{Z}/f(x_i)\mathbb{Z})-\{0\}$ for all $i\in\{1,\ldots,k\}$, and $ b_j\in\mathbb{Z}-\{0\}$ for all $j\in\{1,\ldots,l\}$. Notice that such an element always exists. It is easily seen that $l_{sy}(gx_i^{-a_i}) \lt l_{sy}(g)$ for all $i\in\{1,\ldots,k\}$ and $l_{sy}(gy_j^{-b_j}) \lt l_{sy}(g)$ for all $j\in\{1,\ldots,l\}$. On the other hand, if $x\in V_2$, then $l(w_0x) \lt l(w_0)$, hence $l_{sy}(gx) \lt l_{sy}(g)$. So, $g\in B_\emptyset$.

Let (D, V) be a Dyer system of spherical type. So, $D=D_2\times D_p\times D_\infty$, D ₂ is a finite Coxeter group, $D_p=\prod_{x\in V_p}\mathbb{Z}/f(x)\mathbb {Z}$, and $D_\infty=\mathbb{Z}^l$, where $l=|V_\infty|$. Let $x\in V_p$. If $f(x)=2r$ is even we set $P_x(t)=2t+2t^2+\ldots+2t^{r-1}+t^r$, and if $f(x)=2r+1$ is odd we set $P_x(t)=2t+2t^2+\ldots+2t^r$. Then we set

\begin{equation*}P_D(t)=t^m\left(\prod_{x\in V_p}P_x(t)\right)\frac{2^lt^l}{(1-t)^l},\end{equation*}

where m is the maximal length in D ₂.

As an immediate corollary we obtain the following whose first part finishes the proof of Theorem 1.1.

Proof. By Proposition 4.6 it suffices to show that $\mathcal{G}_{(B_\emptyset,V)}(t)=0$ if D is not of spherical type and that $\mathcal {G}_{(B_\emptyset,V)}(t)=P_D(t)$ if D is of spherical type. If D is not of spherical type then, by Lemma 4.10, $B_\emptyset=\emptyset$, hence $\mathcal{G}_{(B_\emptyset,V)}(t)=0$. Suppose D is of spherical type. Let $V_p=\{x_1,\dots,x_k\}$ and $V_\infty=\{y_1,\dots,y_l\}$, and let m be the maximal length in D ₂. Then, by Lemma 4.10,

\begin{gather*} \mathcal{G}_{(B_\emptyset,V)}(t)=t^m\left(\prod_{i=1}^k \mathcal{G}_{(\mathbb{Z}/f(x_i)\mathbb{Z}-\{0\},\{1\})}(t) \right)\left(\prod_{i=1}^l\mathcal{G}_{(\mathbb{Z}-\{0\},\{1\})}(t)\right)=\\ t^m\left(\prod_{i=1}^kP_{x_i}(t) \right)\left(\frac{2t}{1-t}\right)^l=P_D(t). \end{gather*}

We end this chapter with the proof of Theorem 1.1.

Proof of Theorem 1.1

The first part follows from Corollary 4.11 which we already mentioned above. We assume now that D is of spherical type, then $D=D_2\times D_p\times D_\infty$. Hence we can use the formula for direct products on page 5. We get

\begin{equation*}\mathcal{G}_{(D,V)}(t)=\mathcal{G}_{(D_2,V_2)}(t)\cdot \mathcal{G}_{(D_p,V_p)}(t)\cdot \mathcal{G}_{(D_\infty,V_\infty)}(t).\end{equation*}

Further, since $\mathcal{G}_{(\mathbb{Z},\left\{1\right\})}(t)=\frac{1+t}{1-t}$ we get $\mathcal{G}_{(D_\infty,V_\infty)}(t)=\frac{(1+t)^l}{(1-t)^l}$ where l is the cardinality of $V_\infty$. A direct calculation shows the formulas for the spherical growth series of finite cyclic groups with the standard generating sets which ends the proof of the second part of the theorem.

5. Euler characteristic

We start this chapter by recalling the definition and useful formulas of the Euler characteristic of groups. Following [Reference Brown3] a group G is said to be of finite homological type if the virtual cohomological dimension of G is finite and for every G-module M which is finitely generated as an abelian group, $H_i(G,M)$ is finitely generated for all i. If G is torsion-free and of finite homological type, then its Euler characteristic is defined by:

\begin{equation*}\chi(G):=\sum (-1)^i rk_{\mathbb{Z}}(H_i(G)).\end{equation*}

If G is of finite homological type and has a torsion free subgroup H of finite index, then the Euler characteristic of G is defined by:

\begin{equation*}\chi(G):=\frac{\chi(H)}{[G:H]}.\end{equation*}

We list some useful properties of the Euler characteristic.

As a corollary we obtain

Theorem 5.3 Let $(\Gamma, m,f)$ be a Dyer graph and (D, V) be the associated Dyer system. Then

\begin{equation*}\frac{1}{\mathcal{G}_{(D,V)}(1)}=\chi(D).\end{equation*}

Proof. Assume first that Γ is complete. In this case $D=D_2\times D_p\times D_\infty$, where D_p is finite and $D_\infty\cong\mathbb{Z}^l$. Hence, by Corollary 5.2

\begin{equation*}\chi(D)=\chi(D_2)\cdot\chi(D_p)\cdot\chi(\mathbb{Z}^l).\end{equation*}

Since D ₂ is a Coxeter group we know by [Reference Serre15] that $\chi(D_2)=\frac{1}{\mathcal{G}_{(D_2, V_2)}(1)}$. We get

\begin{equation*}\chi(D)=\frac{1}{\mathcal{G}_{(D_2, V_2)}(1)}\cdot\chi(D_p)\cdot\chi(\mathbb{Z}^l).\end{equation*}

Further, since D_p is finite we have $\chi(D_p)=\frac{1}{\mid D_p\mid}$ and $\mathcal{G}_{(D_p, V_p)}(1)=|D_p|$. Thus

\begin{equation*}\chi(D)=\frac{1}{\mathcal{G}_{(D_2, V_2)}(1)}\cdot\frac{1}{\mathcal{G}_{(D_p, V_p)}(1)}\cdot\chi(\mathbb{Z}^l).\end{equation*}

We know that $\chi(\mathbb{Z}^l)=0$ and $\frac{1}{\mathcal{G}_{(D_\infty, V_\infty)}(1)}=\frac{(1-1)^l}{(1+1)^l}=0$ if l > 0, and $\chi(\mathbb{Z}^l)=1$ and $\frac{1}{\mathcal{G}_{(D_\infty, V_\infty)}(1)}=1$ if l = 0. Hence

\begin{equation*}\chi(D)=\frac{1}{\mathcal{G}_{(D_2, V_2)}(1)}\cdot\frac{1}{\mathcal{G}_{(D_p, V_p)}(1)}\cdot\frac{1}{\mathcal{G}_{(D_\infty, V_\infty)}(1)}=\frac{1}{\mathcal{G}_{(D,V)}(1)}.\end{equation*}

Now assume that Γ is not complete, then there exists $x\in V(\Gamma)$ such that $st(x)\neq V(\Gamma)$. Then we have

\begin{equation*}D=D_{V-\left\{x\right\}}*_{D_{lk(x)}}D_{st(x)}.\end{equation*}

By Corollary 3.7 we obtain

\begin{equation*}\frac{1}{\mathcal{G}_{(D, V)}(1)}=\frac{1}{\mathcal{G}_{(D_{V-\{x\}}, V-\{x\})}(1)}+\frac{1}{\mathcal{G}_{(D_{st(x)}, st(x))}(1)}-\frac{1}{\mathcal{G}_{(D_{lk(x)}, lk(x))}(1)}.\end{equation*}

By Corollary 5.2 we get

\begin{equation*}\chi(D)=\chi(D_{V-\left\{x\right\}}*_{D_{lk(x)}}D_{st(x)})=\chi(D_{V-\{x\}})+\chi(D_{st(x)})-\chi(D_{lk(x)}).\end{equation*}

We decompose $D_{V-\left\{x\right\}}$, $D_{st(x)}$ and $D_{lk(x)}$ again in amalgamated products. Using this strategy we will get a linear combination of Euler characteristics resp. spherical growth series of standard parabolic subgroups of D where all defining graphs are complete. Applying the above formulas we get

\begin{equation*}\frac{1}{\mathcal{G}_{(D,V)}(1)}=\chi(D).\end{equation*}