2.1 Totally disconnected locally compact (tdlc) groups

Let G be a tdlc group. It is a classical result by Van Dantzig that the set of all compact open subgroups of G forms a basis of neighborhoods of the identity. The scale function s of G is defined as follows:

$$\begin{align*}s: G \to \mathbb{N} : g \mapsto \min \{ [ gUg^{-1} : gUg^{-1} \cap U ] \mid U\leq G \text{ compact and open} \}. \end{align*}$$

Let $g \in G$ and let $U \leq G$ be a compact open subgroup. Then U is said to be minimizing for g if $s(g) = [ gUg^{-1} : gUg^{-1} \cap U ]$ . We say that U is tidy for g if the following two conditions are satisfied:

1. (TA) $U = U_{g +} U_{g -}$ , where $U_{g+} := \bigcap _{n\geq 0} g^n U g^{-n}$ and $U_{g-} := \bigcap _{n\geq 0} g^{-n} U g^n$ ;

2. (TB) $\widehat {U}_{g +} := \bigcup _{n\geq 0} g^n U_{g +} g^{-n}$ and $\widehat {U}_{g -} := \bigcup _{n\geq 0} g^{-n} U_{g -} g^n$ are closed in G.

A compact open subgroup is said to be tidy above for g if it satisfies (TA), and tidy below for g if it satisfies (TB). The next Theorem shows the connection between minimizing and tidy subgroups.

The following results are useful in this paper.

Let $g\in G$ . The contraction group of g is defined as follows:

$$\begin{align*}\mathrm{con}_G(g) := \{ h\in G \mid \lim_{n \to \infty} g^n h g^{-n} = 1_G \}. \end{align*}$$

It is well-known that for $g, h \in G$ and $n\in \mathbb {N}^*$ we have $\mathrm {con}(g) = \mathrm {con}(g^n)$ and $h\mathrm {con}(g)h^{-1} = \mathrm {con}( hgh^{-1} )$ . It follows from [Reference WillisWil14, Remark 3.1(d)] that the intersection of all subgroups being tidy for g coincides with the intersection of the closures of the contraction groups of g and $g^{-1}$ . This common subgroup is called the nub of g and is denoted by

$$\begin{align*}\mathrm{nub}_G(g) := \bigcap \{ U \leq G \mid U \text{ is tidy for } g \} = \overline{\mathrm{con}_G(g)} \cap \overline{\mathrm{con}_G(g^{-1})}. \end{align*}$$

We simply write $\mathrm {con}(g)$ and $\mathrm {nub}(g)$ , if the group G is clear from the context. For $g\in G$ we call a subset $X \subseteq G$ g-stable if $gXg^{-1} = X$ .

2.2 Coxeter systems

Let $(W, S)$ be a Coxeter system, which we will always assume to be of finite rank, that is, with S finite. We shall denote by $\ell =\ell _S$ the corresponding word metric with respect to S.

It is well-known that for each $J \subseteq S$ the pair $(W_J := \langle J \rangle , J)$ is a Coxeter system (cf. [Reference BourbakiBou02, Ch. IV, § $1$ Theorem $2$ ]). For $s, t \in S$ we denote the order of $st$ in W by $m_{st}$ . The Coxeter diagram corresponding to $(W, S)$ is the labeled graph $(S, E(S))$ , where $E(S) = \{ \{s, t \} \mid m_{st}>2 \}$ and where each edge $\{s,t\}$ is labeled by $m_{st}$ for all $s, t \in S$ . The Coxeter system $(W, S)$ is called irreducible if the underlying Coxeter diagram is connected. We say that $J \subseteq S$ is irreducible if $(W_J, J)$ is irreducible; J is called a component of S if it is the vertex set of a connected component of the Coxeter diagram corresponding to $(W, S)$ .

A subset $J \subseteq S$ is called spherical if $W_J$ is finite. The Coxeter system is called spherical if S is spherical, and of finite type if in addition it is irreducible. If $(W, S)$ is irreducible, then $(W, S)$ (resp. S) is said to be of affine type if W is infinite and virtually abelian; equivalently, its underlying Coxeter diagram is of type $\tilde {A}_n$ , $\tilde {B}_n$ , $\tilde {C}_n$ , $\tilde {D}_n$ , $\tilde {E}_6$ , $\tilde {E}_7$ , $\tilde {E}_8$ , $\tilde {F}_4$ or $\tilde {G_2}$ (see [Reference BourbakiBou02, Ch. VI, $\S 4$ Theorem $4$ ]). The irreducible Coxeter system $(W, S)$ (resp. S) is said to be of indefinite type if it is neither spherical nor affine. For $J \subseteq S$ we let $J_{\mathrm {aff}}^{\mathrm {Cox}} \subseteq J$ denote the union of all the components of J of affine type. We call $J \subseteq S$ essential if J is nonempty and has no spherical component. We further set $J^{\perp } := \{ s\in S \mid m_{sj} = 2 \text { for all } j \in J \}$ .

The subgroups of W of the form $W_J$ for some $J\subseteq S$ are called standard parabolic subgroups, and their conjugates parabolic subgroups. A parabolic subgroup is called spherical if it is finite. The intersection $\mathrm {Pc}(w)$ of all parabolic subgroups containing an element $w \in W$ is again a parabolic subgroup, called the parabolic closure of w. A nontrivial element $w\in W$ is called straight if $\ell (w^n)=n\ell (w)$ for all $n\in \mathbb {N}$ . An element $w\in W$ is called cyclically reduced if $\ell (w) = \min \{ \ell (vwv^{-1}) \mid v\in W \}$ . We note that straight elements are cyclically reduced by [Reference MarquisMar14b, Lemma $4.1$ ]. For $w\in W$ , a decomposition $w = s_1 \cdots s_n$ with $s_1, \ldots , s_n \in S$ and $\ell (w) = n$ is called reduced. The support $\mathrm {supp}(w)$ of $w\in W$ is the subset J of elements in S appearing in some (equivalently, any) reduced decomposition of w.

2.3 Standard elements

Suppose $(W, S)$ is of irreducible indefinite type, and let $w\in W$ with $\mathrm {Pc}(w)=W$ . By [Reference MarquisMar25, Theorem C $(1)$ ], there exists a largest spherical parabolic subgroup $P_w^{\max }$ which is normalized by w, and we have $P_w^{\max }=P_{w^n}^{\max }$ for all $n\in \mathbb {N}^*$ . Following [Reference MarquisMar25], the element $w\in W$ is called standard if it is cyclically reduced and $P_w^{\max }$ is standard.

Back to a general Coxeter system $(W, S)$ , let $w\in W$ be such that $\mathrm {Pc}(w)=W_J$ for some essential $J\subseteq S$ . Let $J_1, \ldots , J_k$ be the components of J of indefinite type and let $J_0=J_{\mathrm {aff}}^{\mathrm {Cox}}$ , so that $J=J_0\cup J_1\cup \cdots \cup J_k$ . Let $w_i \in W_{J_i}$ such that $w = w_0 \cdots w_k$ . We call w standard if $w_i$ is standard for each $i= 1,\ldots ,k$ . In this case we let $I_i \subseteq J_i$ with $P_{w_i}^{\max } = W_{I_i}$ for all $i= 1,\ldots ,k$ and define $J_{\mathrm {sph}}^w := I_1 \cup \cdots \cup I_k$ .

2.4 Buildings

A building of type (W, S) is a pair $\mathcal {B} = (\mathcal {C}, \delta )$ where $\mathcal {C}$ is a nonempty set and where $\delta : \mathcal {C} \times \mathcal {C} \to W$ is a distance function satisfying the following axioms, where $x, y\in \mathcal {C}$ and $w = \delta (x, y)$ :

1. (Bu1) $w = 1_W$ if and only if $x=y$ ;

2. (Bu2) if $z\in \mathcal {C}$ satisfies $s := \delta (y, z) \in S$ then $\delta (x, z) \in \{w, ws\}$ , and if, furthermore, $\ell (ws) = \ell (w) +1$ then $\delta (x, z) = ws$ ;

3. (Bu3) if $s\in S$ , there exists $z\in \mathcal {C}$ such that $\delta (y, z) = s$ and $\delta (x, z) = ws$ .

The elements of $\mathcal {C}$ are called chambers. Two chambers $x, y \in \mathcal {C}$ are called adjacent if $\delta (x, y) = s$ for some $s\in S$ (in which case x and y are said to be s-adjacent). A gallery from x to y is a sequence of chambers $(x = x_0, \ldots , x_k = y)$ such that for each $1 \leq l \leq k$ , the chambers $x_{l-1}$ and $x_l$ are $s_l$ -adjacent for some $s_l\in S$ . The number k is then called the length of the gallery and the tuple $(s_1,\ldots ,s_k)$ its type. A gallery from x to y of length k is called minimal if there is no gallery from x to y of length $<k$ ; in this case we write ${d_{\mathrm {Ch}}}(x, y) = k$ . Assigning to each gallery its type yields a 1–1 correspondence between minimal galleries from x to y and reduced decompositions of $w=\delta (x,y)$ (cf. [Reference Abramenko and BrownAB08, Exercise 5.20]); in particular, ${d_{\mathrm {Ch}}}(x, y) = \ell (\delta (x, y))$ for all $x, y \in \mathcal {C}$ .

2.5 Projections and residues

Let $\mathcal {B} = (\mathcal {C}, \delta )$ be a building of type $(W, S)$ . Given a subset $J \subseteq S$ and $x\in \mathcal {C}$ , the J-residue of x is the set $R_J(x) := \{y \in \mathcal {C} \mid \delta (x, y) \in W_J \}$ . Each J-residue is a building of type $(W_J, J)$ with the distance function induced by $\delta $ (cf. [Reference Abramenko and BrownAB08, Corollary $5.30$ ]). A residue is a subset R of $\mathcal {C}$ such that there exist $J \subseteq S$ and $x\in \mathcal {C}$ with $R = R_J(x)$ . The subset J is uniquely determined by R and is called the type of R; the cardinality of J is the rank of R. A residue is called spherical if its type is a spherical subset of S. A panel is a residue of rank $1$ . An s-panel is a panel of type $\{s\}$ for $s\in S$ . For $c\in \mathcal {C}$ and $s\in S$ we put $\mathcal {P}_s(c) := R_{\{s\}}(c)$ . The building $\mathcal {B}$ is called thick, if each panel of $\mathcal {B}$ contains at least three chambers.

Given $x\in \mathcal {C}$ and a residue $R \subseteq \mathcal {C}$ , there exists a unique chamber $z\in R$ such that ${d_{\mathrm {Ch}}}(x, y) = {d_{\mathrm {Ch}}}(x, z) + {d_{\mathrm {Ch}}}(z, y)$ for each $y\in R$ (cf. [Reference Abramenko and BrownAB08, Proposition 5.34]). The chamber z is called the projection of x onto R and is denoted by $\mathrm {proj}_R x$ . Moreover, if $z = \mathrm {proj}_R x$ we have $\delta (x, y) = \delta (x, z) \delta (z, y)$ for each $y\in R$ .

For two residues R and T of $\mathcal {B}$ we define $\mathrm {proj}_T R := \{ \mathrm {proj}_T r \mid r \in R \}$ . The residues R and T are called parallel if $\mathrm {proj}_T R = T$ and $\mathrm {proj}_R T = R$ . In this case, the element $\delta (r,\mathrm {proj}_Tr)\in W$ is independent of the choice of $r\in R$ and is denoted $\delta (R,T)$ . Let $R_J$ and $R_K$ be two residues of $\mathcal {B}$ of type J and K, respectively. Then $\mathrm {proj}_{R_K} R_J$ is again a residue (contained in $R_K$ ) and is of type $K \cap w^{-1} J w$ for some $w\in W$ . If, additionally, $R_J$ and $R_K$ are parallel, then $K = w^{-1} J w$ for some $w\in W$ (cf. [Reference Abramenko and BrownAB08, Proposition 5.37(1)]).

A subset $\Sigma \subseteq \mathcal {C}$ is called convex if for any two chambers $c, d \in \Sigma $ and any minimal gallery $(c_0 = c, \ldots , c_k = d)$ we have $c_i \in \Sigma $ for all $i=0,\ldots ,k$ . We note that residues are convex (cf. [Reference Abramenko and BrownAB08, Example 5.44(b)]). We denote the convex hull of a set $X \subseteq \mathcal {C}$ by $\operatorname {\mathrm {conv}}(X)$ . A subset $\Sigma \subseteq \mathcal {C}$ is called thin if $P \cap \Sigma $ contains exactly two chambers for each panel $P \subseteq \mathcal {C}$ which meets $\Sigma $ . An apartment is a nonempty subset $\Sigma \subseteq \mathcal {C}$ , which is convex and thin.

2.6 Automorphisms

Let $\mathcal {B} = (\mathcal {C}, \delta )$ and $\mathcal {B}' = (\mathcal {C}', \delta ')$ be two buildings of type $(W, S)$ . An isometry between $\mathcal {X} \subseteq \mathcal {C}$ and $\mathcal {X}' \subseteq \mathcal {C}'$ is a bijection $\varphi : \mathcal {X} \to \mathcal {X}'$ such that $\delta '(\varphi (x), \varphi (y)) = \delta (x, y)$ for all $x, y \in \mathcal {X}$ . In this case $\mathcal {X}$ and $\mathcal {X}'$ are called isometric. An isometry $\varphi : \mathcal {C} \to \mathcal {C}$ is called a type-preserving automorphism of $\mathcal {B}$ . We denote their set by $\mathrm {Aut}_0(\mathcal {B})$ .

Let G be a group acting on $\mathcal {B} = (\mathcal {C}, \delta )$ by type-preserving automorphisms. We say that this action is Weyl-transitive if for each $w\in W$ , the group G acts transitively on the set of ordered pairs $(c, d) \in \mathcal {C} \times \mathcal {C}$ with $\delta (c, d) = w$ .

2.7 Roots

A reflection is an element of W that is conjugate to an element of S. For $s\in S$ we let $\alpha _s := \{ w\in W \mid \ell (sw)> \ell (w) \}$ be the simple root corresponding to s. A root is a subset $\alpha \subseteq W$ such that $\alpha = v\alpha _s$ for some $v\in W$ and $s\in S$ . We denote the set of all roots by $\Phi := \Phi (W, S)$ . The set $\Phi _+ = \{ \alpha \in \Phi \mid 1_W \in \alpha \}$ is the set of all positive roots and $\Phi _- = \{ \alpha \in \Phi \mid 1_W \notin \alpha \}$ is the set of all negative roots. For each root $\alpha \in \Phi $ we set $-\alpha := W \setminus \alpha \in \Phi $ and we denote the unique reflection interchanging $\alpha $ and $-\alpha $ by $r_{\alpha } \in W \leq \mathrm {Aut}_0(\Sigma (W, S))$ : if $\alpha = v\alpha _s$ for some $v\in W$ and $s\in S$ , then $r_{\alpha }=vsv^{-1}$ . For a root $\alpha \in \Phi $ we define the support of $\alpha $ as $\mathrm {supp}(\alpha ) := \mathrm {supp}(r_{\alpha })$ . For $\alpha \in \Phi $ we denote by $\partial \alpha $ the set of all panels of $\Sigma (W, S)$ stabilized by $r_{\alpha }$ and call it the wall associated with $\alpha $ .

For a subset $Z \subseteq W$ we denote by $\operatorname {\mathrm {conv}}(Z)$ its convex hull (with respect to the chamber distance ${d_{\mathrm {Ch}}}$ ); equivalently, $\operatorname {\mathrm {conv}}(Z)$ is the intersection of all roots containing Z (see [Reference Abramenko and BrownAB08, Proposition 3.94]).

2.8 RGD systems

An RGD system of type (W, S) is a pair $\mathcal {D} = \left ( G, \left ( U_{\alpha } \right )_{\alpha \in \Phi }\right )$ consisting of a group G together with a family of subgroups $U_{\alpha }$ (called root groups) indexed by the set of roots $\Phi = \Phi (W, S)$ , satisfying some axioms (for the precise axioms we refer the reader to [Reference Abramenko and BrownAB08, Sections 7.8 and 8.6]). We will only need certain properties of RGD systems which we now discuss.

Let $\mathcal {D} = (G, (U_{\alpha })_{\alpha \in \Phi })$ be an RGD system of type $(W, S)$ . We set $T := \bigcap _{\alpha \in \Phi } N_G(U_{\alpha })$ , $U_{\pm } := \langle U_{\alpha } \mid \alpha \in \Phi _{\pm } \rangle $ and $B := T U_+ = T \ltimes U_+$ . There exists a subgroup N of G normalizing T such that $N/T \cong W$ and such that $(B,N)$ is a BN-pair for G (see [Reference Abramenko and BrownAB08, Section 6.2.6]) – when no confusion is possible, we will often make the slight abuse of notation of viewing W as a subgroup of G. In particular, there exists a thick building $\mathcal {B}=\mathcal {B}(\mathcal {D})$ with chamber set $G/B$ on which G acts Weyl-transitively, by left multiplication (see [Reference Abramenko and BrownAB08, Chapter $8$ ] for more information). From now on we will relax notation and also denote by $\mathcal {B}$ the set of chambers of $\mathcal {B}$ . The trivial coset $c:=1_GB$ in $G/B$ is called the fundamental chamber (thus $B = \mathrm {Fix}_G(c)$ ). The fundamental apartment $\Sigma :=\{ wc \ | w\in W\}\subseteq \mathcal {B}$ of $\mathcal {B}$ can be W-equivariantly identified with $\Sigma (W,S)$ (with $c\in \Sigma $ corresponding to $1_W\in \Sigma (W,S)$ ), and we have $N = \mathrm {Stab}_G(\Sigma )$ and $T = \mathrm {Fix}_G(\Sigma )$ . We will then also refer to the roots of $(W,S)$ as roots of $\Sigma $ , and keep the notation $\Phi $ to denote their set. The action of the root groups on $\mathcal {B}$ has the following properties (cf. [Reference Abramenko and BrownAB08, Section $\mathrm{8.6.4}$ ]), where $\alpha \in \Phi $ :

• ○ $U_{\alpha }$ fixes the root $\alpha $ of $\Sigma $ as well as each interior panel P of $\alpha $ , that is, each panel P with the property $\vert P \cap \alpha \vert = 2$ .

• ○ For each boundary panel P of $\alpha $ (i.e., a panel P with the property $\vert P \cap \alpha \vert = 1$ ) the action of the root group on $P \setminus \{p\}$ is simply transitive, where p is the unique chamber contained in $P \cap \alpha $ .

5.1 Tidiness of V

We want to show that for $w\in W$ straight the subgroup V is tidy for $\widetilde {w}$ . To do so we first show that $H_c$ is tidy and deduce that V is minimizing.

As G acts Weyl-transitively on $\mathcal {B}$ , for each $s\in S$ , all s-panels of $\mathcal {B}$ have the same cardinality. Moreover, as all root groups are finite and act simply transitively on panels, all panels are finite. If P is an s-panel, we set $q_s := \vert P \vert -1=|U_{\alpha _s}|$ .

The following lemma is analogous to [Reference Baumgartner, Parkinson and RamaggeBPR19, Theorem 4.7] and its proof follows essentially the same lines. However, in [Reference Baumgartner, Parkinson and RamaggeBPR19] the completion H is assumed to be a subgroup of $\mathrm {Aut}_0(\mathcal {B})$ . Moreover, they show that the stabilizer of a suitable spherical residue is tidy, while we show that the stabilizer of a chamber is tidy.

Proof. Lemma 2.2 $(1,2)$ implies that $H_c$ is tidy for $\widetilde {w}$ (equivalently, for $\widetilde {w}^{-1}$ ) if and only if

$$\begin{align*}[H_c: H_c \cap H_{\widetilde{w}^nc}] = [H_c: H_c \cap \widetilde{w}^n H_c \widetilde{w}^{-n}] \overset{!}{=} [H_c: H_c \cap \widetilde{w} H_c \widetilde{w}^{-1}]^n = [H_c: H_c \cap H_{\widetilde{w}c}]^n \end{align*}$$

holds for all $n\in \mathbb {N}$ . Let $(c_0 := c, \ldots , c_k := \widetilde {w}c)$ be a minimal gallery. As w is straight, the concatenation of the minimal galleries $(\widetilde {w}^ic_0, \ldots , \widetilde {w}^ic_k)$ , $0 \leq i \leq n-1$ , is a minimal gallery from c to $\widetilde {w}^nc$ , which we denote $(d_0, \ldots , d_{kn})$ . By Lemma 2.9,

$$ \begin{align*} \left[ H_c: H_c \cap H_{\widetilde{w}^nc} \right] = \left[ H_{d_0}: \bigcap\limits_{i=0}^{kn} H_{d_i} \right] = \prod_{i=1}^{kn} \left[ \bigcap\limits_{j=0}^{i-1} H_{d_j} : \bigcap\limits_{j=0}^i H_{d_j} \right]. \end{align*} $$

Let $P_i$ be the panel containing $d_{i-1}$ and $d_i$ and let $s_i \in S$ be the type of $P_i$ . Then $d_{i-1} = \mathrm {proj}_{P_i} d_0$ .

We claim that $\left [ \bigcap _{j=0}^{i-1} H_{d_j}: \bigcap _{j=0}^i H_{d_j} \right ] =q_{s_i}$ . Indeed, let $p \in P_i\setminus \{d_{i-1}\}$ . As $\delta (d_0, d_i) = \delta (d_0, d_{i-1}) s_i = \delta (d_0, p)$ and as H acts Weyl-transitively on $\mathcal {B}$ , there exists $h=h_p\in H_{d_0}$ with $hd_i = p$ . Lemma 2.9 implies that $hd_{i-1} = h \left ( \mathrm {proj}_{P_i} d_0 \right ) = \mathrm {proj}_{hP_i} (hd_0) = \mathrm {proj}_{P_i} d_0 = d_{i-1}$ . Again by Lemma 2.9 we deduce that $h\in \bigcap _{j=0}^{i-1} H_{d_j}$ . Note, moreover, that if $p' \in P_i\setminus \{d_{i-1}\}$ is such that $h_{p'}H_{d_i}=h_pH_{d_i}$ , then $p=p'$ . Thus $\{h_p \ | \ p \in P_i\setminus \{d_{i-1}\}\}$ is a set of coset representatives for $\bigcap _{j=0}^i H_{d_j}$ in $\bigcap _{j=0}^{i-1} H_{d_j}$ , yielding the claim.

We conclude that

$$ \begin{align*} \prod_{i=1}^{kn} \left[ \bigcap_{j=0}^{i-1} H_{d_j}: \bigcap_{j=0}^i H_{d_j} \right] &= \prod_{i=1}^{kn} q_{s_i} = \left( q_{s_1} \cdots q_{s_k} \right)^n = \left[ H_c: H_c \cap H_{\widetilde{w}c} \right]^n. \end{align*} $$

Note that the proof also shows that $s(\widetilde {w}^{-1}) = [H_c : H_c \cap H_{\widetilde {w}c}] = q_{s_1} \cdots q_{s_k}$ , where $(s_1, \ldots , s_k)$ is the type of a minimal gallery from c to $\widetilde {w}c$ . We deduce that $s(\widetilde {w}) = q_{s_k} \cdots q_{s_1} = q_{s_1} \cdots q_{s_k}$ .

Proof. By Lemma 5.1 we know that $H_c$ is tidy for $\widetilde {w}$ and $s(\widetilde {w}) = q_{s_1} \cdots q_{s_k}$ , where $w=s_1\ldots s_k$ is a reduced decomposition of w. To show that V is tidy for $\widetilde {w}$ (equivalently, for $\widetilde {w}^{-1}$ ) it thus suffices by Theorem 2.1 to show that

$$ \begin{align*}[V : V\cap \widetilde{w} V \widetilde{w}^{-1}]=[\widetilde{w}^{-1} V \widetilde{w} : \widetilde{w}^{-1} V \widetilde{w} \cap V] \leq s(\widetilde{w}^{-1}) = q_{s_1} \cdots q_{s_k}.\end{align*} $$

By Lemma 3.4 we have $V \cap H_{\widetilde {w}c} = V \cap \widetilde {w} V \widetilde {w}^{-1}$ . To conclude, note that the map

$$ \begin{align*}V/(V \cap H_{\widetilde{w}c})\to \{ d\in \mathcal{B} \mid \delta(c, d) = \delta(c, \widetilde{w}c) \}: h(V \cap H_{\widetilde{w}c})\mapsto h\widetilde{w}c\end{align*} $$

is injective, and that $\vert \{ d\in \mathcal {B} \mid \delta (c, d) = \delta (c, \widetilde {w}c) \} \vert = q_{s_1} \cdots q_{s_k}$ . The second assertion follows from Lemma 2.2(3).

5.2 Upper bound for the nub

The purpose of this subsection is to give an upper bound for the nub of an element $\widetilde {w}$ with $w\in W$ straight.

Theorem 5.6. Let $w\in W$ be straight and standard, and let $J = \mathrm {supp}(w)$ . Then:

$$\begin{align*}\mathrm{nub}(\widetilde{w}) \leq \left( \bigcap_{z\in \mathbb{Z}} \widetilde{w}^z V \widetilde{w}^{-z} \right) \cap \mathrm{Fix}_H( R_{J^{\perp} \cup J_{\mathrm{sph}}^w}(c) ) \cap \mathrm{Fix}_H( \Sigma \cap R_{J_{\mathrm{aff}}^{\mathrm{Cox}}}(c) ). \end{align*}$$

5.3 Contraction of root groups

We next provide a lower bound for the nub of an element $\widetilde {w}$ with $w\in W$ straight and standard.

6.1 Preliminaries on Kac–Moody algebras and groups

Generalized Cartan matrices

Throughout, we fix a generalized Cartan matrix (GCM) $A=(a_{ij})_{i,j\in I}$ , as well as a realization $(\mathfrak {h},\Pi ,\Pi ^{\vee })$ of A in the sense of [Reference KacKac90, §1.1], with set of simple roots $\Pi =\{\alpha _i \ | \ i\in I\}$ and set of simple coroots $\Pi ^{\vee }=\{\alpha ^{\vee }_i \ | \ i\in I\}$ .

The Weyl group $W = W(A)$ of A is the subgroup of $\operatorname {\mathrm {GL}}(\mathfrak {h}^*)$ generated by the simple reflections $s_i$ ( $i\in I$ ) defined by

$$ \begin{align*}s_i\colon\thinspace\mathfrak{h}^*\to\mathfrak{h}^*: \alpha\mapsto \alpha-\langle\alpha,\alpha_i^{\vee}\rangle\alpha_i.\end{align*} $$

The pair $(W, S:=\{s_i \ | \ i\in I\})$ is then a Coxeter system. When convenient, we will identify a subset $J\subseteq I$ with $\{s_i \ | \ i\in J\}\subseteq S$ (writing for instance $W_J$ for the parabolic subgroup generated by $\{s_i \ | \ i\in J\}$ ).

We call A indecomposable if there is no proper nonempty $J\subseteq I$ with $a_{ij}=0$ for all $i\in I\setminus J$ and $j\in J$ , or equivalently if $(W, S)$ is irreducible. An indecomposable GCM A can be of finite, affine or indefinite type (see [Reference KacKac90, Chapter 4] or Remark 6.1 below). Accordingly, we call a subset $J\subseteq I$ of finite, affine or indefinite type if the GCM $A_J:=(a_{ij})_{i,j\in J}$ is of that type. As before, we define the components of $J\subseteq I$ as the subsets $J_1,\ldots ,J_r\subseteq J$ such that $W_J=W_{J_1}\times \cdots \times W_{J_r}$ with $J_i$ irreducible for each i. Given $J\subseteq I$ , we denote by $J_{\mathrm {aff}}$ the union of all the components of J of affine type.

Kac–Moody root systems

The Kac–Moody algebra $\mathfrak {g}(A)$ associated with A admits a root space decomposition $\mathfrak {g}(A)=\mathfrak {h}\oplus \bigoplus _{\alpha \in \Delta }\mathfrak {g}_{\alpha }$ with respect to the adjoint action of the Cartan subalgebra $\mathfrak {h}$ , with corresponding root spaces

$$ \begin{align*}\mathfrak{g}_{\alpha}:=\{x\in\mathfrak{g}(A) \ | \ [h,x]=\alpha(h)x \ \forall h\in\mathfrak{h}\}\end{align*} $$

and root system $\Delta :=\{\alpha \in \mathfrak {h}^*\setminus \{0\} \ | \ \mathfrak {g}_{\alpha }\neq \{0\}\}$ . We also set $\mathfrak {g}_A:=[\mathfrak {g}(A),\mathfrak {g}(A)]=\mathfrak {h}'\oplus \bigoplus _{\alpha \in \Delta }\mathfrak {g}_{\alpha }$ , where $\mathfrak {h}':=\sum _{i\in I}\mathbb {C}\alpha _i^{\vee }\subseteq \mathfrak {h}$ .

Set $Q_+:=\bigoplus _{i\in I}\mathbb {N}\alpha _i$ . We let $\Delta ^+:=\Delta \cap Q_+$ denote the set of positive roots, so that $\Delta =\Delta ^+\dot {\cup }(-\Delta ^+)$ . The height of a root $\alpha =\pm \sum _{i\in I}n_i\alpha _i$ is $\operatorname {\mathrm {ht}}(\alpha ):=\pm \sum _{i\in I}n_i\in \mathbb {Z}$ . We also define the support

$$ \begin{align*}\mathrm{supp}(\alpha):=\{i\in I \ | \ n_i\neq 0\}\subseteq I\end{align*} $$

of an element $\alpha \in \pm Q_+$ (see also Remark 6.2). The support of a root is always connected. For a subset $J\subseteq I$ , we set

(6.1) $$ \begin{align} \Delta(J):=\Delta\cap \bigoplus_{i\in J}\mathbb{Z}\alpha_i, \quad \Delta^+(J):=\Delta^+\cap\Delta(J), \quad \text{and}\quad \Delta^+_J:=\Delta^+\setminus\Delta^+(J). \end{align} $$

The Weyl group W stabilizes $\Delta $ . One defines the sets of real and imaginary roots, respectively, as

$$ \begin{align*}\Delta^{\operatorname{\mathrm{re}}}:=W.\{\alpha_i \ | \ i\in I\}\subseteq\Delta\quad\text{and}\quad \Delta^{\operatorname{\mathrm{im}}}:=\Delta\setminus\Delta^{\operatorname{\mathrm{re}}}.\end{align*} $$

One also sets $\Delta ^{\operatorname {\mathrm {re+}}}:=\Delta ^{\operatorname {\mathrm {re}}}\cap \Delta ^+$ and $\Delta ^{\operatorname {\mathrm {im+}}}:=\Delta ^{\operatorname {\mathrm {im}}}\cap \Delta ^+$ , as well as $\Delta ^{\operatorname {\mathrm {im+}}}(J) := \Delta ^{\operatorname {\mathrm {im+}}} \cap \Delta (J)$ for each $J \subseteq I$ .

The set of positive imaginary roots can be described as $\Delta ^{\operatorname {\mathrm {im+}}}=\bigcup _{w\in W}wK_0$ , where

$$ \begin{align*}K_0:=\{\alpha\in Q_+ \setminus \{0\} \ | \ \mathrm{supp}(\alpha) \text{ is connected and} \ \langle\alpha,\alpha_i^{\vee}\rangle\leq 0 \ \forall i\in I\}.\end{align*} $$

Moreover, for any $\alpha \in \Delta ^{\operatorname {\mathrm {im+}}}$ , there is a unique root $\alpha _{\min }\in W\alpha \cap K_0$ , namely, $\alpha _{\min }$ is the unique element of minimal height in $W\alpha $ . If $\alpha \in \Delta ^{\operatorname {\mathrm {im+}}}$ , then $\mathrm {supp}(\alpha )$ is of affine or indefinite type. If $\alpha \in K_0$ , then $\mathrm {supp}(\alpha )$ is of affine type if and only if

(6.2) $$ \begin{align} \langle\alpha,\alpha_i^{\vee}\rangle=0 \quad\text{for all } i\in\mathrm{supp}(\alpha). \end{align} $$

Kac–Moody groups

Let $k=\mathbb {F}_q$ be a finite field of order q and characteristic p. Let $\mathfrak {G}_A$ be the constructive Tits functor of simply connected type associated to A introduced by Tits [Reference TitsTit87], and let $G:=\mathfrak {G}_{A}(k)$ be the corresponding minimal Kac–Moody group over k. Thus G is an amalgamated product of a (finite) torus $T:=\mathfrak {T}(k)$ exponentiating $\mathfrak {h}'$ , and of the real root groups $U_{\alpha }:=\mathfrak {U}_{\alpha }(k)\cong (k,+)$ ( $\alpha \in \Delta ^{\operatorname {\mathrm {re}}}$ ) exponentiating the real root spaces $\mathfrak {g}_{\alpha }$ . Then $\mathcal {D}_A := ( G, (U_{\alpha })_{\alpha \in \Delta ^{\operatorname {\mathrm {re}}}} )$ is an RGD system of type $(W,S)$ , and $T=\bigcap _{\alpha \in \Delta ^{\operatorname {\mathrm {re}}}}N_G(U_{\alpha })$ . We keep the notations $U_{\pm }=\langle U_{\alpha } \ | \ \alpha \in \Delta ^{\mathrm {re}\pm }\rangle $ and N from §2.8, and denote as before by $\mathcal {B}=\mathcal {B}(\mathcal {D}_A)$ the associated building, with fundamental chamber c and fundamental apartment $\Sigma \approx \Sigma (W,S)$ . The kernel of the action map $\rho \colon \thinspace G\to \mathrm {Aut}_0(\mathcal {B})$ coincides with the center $\mathcal {Z}(G)\subseteq T$ of G.

We also consider the maximal Kac–Moody group $G^{\operatorname {\mathrm {pma}}}:=\mathfrak {G}_{A}^{\operatorname {\mathrm {pma}}}(k)$ of simply connected type associated to A, which coincides with the scheme-theoretic completion of G (see [Reference MarquisMar18, §8.5]). Thus $G^{\operatorname {\mathrm {pma}}}$ is a first-countable tdlc group constructed as an amalgamated product of G with a certain completion $U^{\operatorname {\mathrm {ma+}}}:=\mathfrak {U}^{\operatorname {\mathrm {ma+}}}_A(k)$ of $U_+$ . The group $U^{\operatorname {\mathrm {ma+}}}$ is topologically generated by the real root groups $U_{\alpha }$ ( $\alpha \in \Delta ^{\operatorname {\mathrm {re+}}}$ ), together with the imaginary root groups $U_{\beta }$ ( $\beta \in \Delta ^{\operatorname {\mathrm {im+}}}$ ) exponentiating the subalgebras $\oplus _{n\in \mathbb {N}^*}\mathfrak {g}_{n\beta }$ .

More precisely, let $\mathfrak {g}_{\mathbb {Z}}=\mathfrak {h}^{\prime }_{\mathbb {Z}}\oplus \bigoplus _{\alpha \in \Delta }\mathfrak {g}_{\alpha \mathbb {Z}}$ be the $\mathbb {Z}$ -form of $\mathfrak {g}_A$ defined in [Reference MarquisMar18, Definition 7.5], and fix a $\mathbb {Z}$ -basis $\mathcal {B}_{\mathbb {Z}}$ of $\mathfrak {n}^+_{\mathbb {Z}}=\bigoplus _{\alpha \in \Delta ^+}\mathfrak {g}_{\alpha \mathbb {Z}}$ consisting of homogeneous elements, that is, of elements $x\in \mathfrak {g}_{\alpha \mathbb {Z}}$ for some $\alpha \in \Delta ^+$ (in which case we set $\deg (x):=\alpha $ ). Choose a total order on $\mathcal {B}_{\mathbb {Z}}$ . Then each element g of $U^{\operatorname {\mathrm {ma+}}}$ can be written in a unique way as a product of “twisted exponentials” $[\exp ](\lambda _xx)$ with $\lambda _x\in k$ and $x\in \mathcal {B}_{\mathbb {Z}}$ (see [Reference MarquisMar18, Theorem 8.51]):

$$ \begin{align*}g=\prod_{x\in\mathcal{B}_{\mathbb{Z}}}[\exp](\lambda_xx)\quad\text{for some } \lambda_x\in k,\end{align*} $$

where the product is taken in the given order on $\mathcal {B}_{\mathbb {Z}}$ (this is called the standard form of g).

More generally, if $\Psi \subseteq \Delta ^+$ is a closed set of roots (i.e., such that $\alpha +\beta \in \Psi $ whenever $\alpha ,\beta \in \Psi $ and $\alpha +\beta \in \Delta $ ), then $\bigoplus _{\alpha \in \Psi }\mathfrak {g}_{\alpha \mathbb {Z}}$ is a subalgebra of $\mathfrak {n}^+_{\mathbb {Z}}$ with $\mathbb {Z}$ -basis $\mathcal {B}_{\Psi }:=\{x\in \mathcal {B}_{\mathbb {Z}} \ | \ \deg (x)\in \Psi \}$ , and the subset $U^{\operatorname {\mathrm {ma}}}_{\Psi } := \mathfrak {U}^{\operatorname {\mathrm {ma}}}_{\Psi }(k)$ of $U^{\operatorname {\mathrm {ma+}}}$ consisting of those elements g of the form $g=\prod _{x\in \mathcal {B}_{\Psi }}[\exp ](\lambda _xx)$ for some $\lambda _x\in k$ is a closed subgroup of $U^{\operatorname {\mathrm {ma+}}}$ . For instance, if $\beta \in \Delta ^{\operatorname {\mathrm {im+}}}$ , then $\Psi =\mathbb {N}^*\delta $ is closed and we set $U_{\beta }:=U^{\operatorname {\mathrm {ma}}}_{\mathbb {N}^*\beta }$ , while if $\alpha \in \Delta ^{\operatorname {\mathrm {re+}}}$ , then $\Psi =\{\alpha \}$ is closed and we have $U_{\alpha }=U^{\operatorname {\mathrm {ma}}}_{\{\alpha \}}$ . In particular, for any closed set $\Psi \subseteq \Delta ^+$ , the group $U^{\operatorname {\mathrm {ma}}}_{\Psi }$ is topologically generated by the root groups $U_{\alpha }$ with $\alpha \in \Psi $ . Note also that for $\Psi _1, \Psi _2 \subseteq \Delta ^+$ closed we have $U^{\operatorname {\mathrm {ma}}}_{\Psi _1} \cap U^{\operatorname {\mathrm {ma}}}_{\Psi _2} = U^{\operatorname {\mathrm {ma}}}_{\Psi _1 \cap \Psi _2}$ .

The group $U^{\operatorname {\mathrm {ma+}}}$ is compact open in $G^{\operatorname {\mathrm {pma}}}$ , and its normal subgroups $U^{\operatorname {\mathrm {ma}}}_n:=\mathfrak {U}^{\operatorname {\mathrm {ma}}}_{\Psi (n)}(k)$ ( $n\in \mathbb {N}$ ), where $\Psi (n):=\{\alpha \in \Delta ^+ \ | \ \operatorname {\mathrm {ht}}(\alpha )\geq n\}$ , form a basis of identity neighborhoods in $G^{\operatorname {\mathrm {pma}}}$ . The torus T normalizes each root group $U_{\alpha }$ ( $\alpha \in \Delta ^{\operatorname {\mathrm {re}}}\cup \Delta ^{\operatorname {\mathrm {im+}}}$ ). The couple $(B^{\operatorname {\mathrm {ma+}}}:=TU^{\operatorname {\mathrm {ma+}}},N)$ is a BN-pair for $G^{\operatorname {\mathrm {pma}}}$ , with the same associated building $\mathcal {B}$ . The action map $G^{\operatorname {\mathrm {pma}}}\to \mathrm {Aut}_0(\mathcal {B})$ is continuous and restricts to $\rho \colon \thinspace G\to \mathrm {Aut}_0(\mathcal {B})$ .

The closure $\overline {G}$ of G in $G^{\operatorname {\mathrm {pma}}}$ coincides with $G^{\operatorname {\mathrm {pma}}}$ as soon as

$$ \begin{align*}p>M_A:=\max_{i\neq j}|a_{ij}|,\end{align*} $$

but may be properly contained in $G^{\operatorname {\mathrm {pma}}}$ otherwise.

Remark 6.5. We fix a section $W \to N: w \mapsto \widetilde {w}$ of the homomorphism $N \to W$ as in [Reference MarquisMar18, Definition $7.58$ ]: for each $i\in I$ , it maps $s_i$ to the element $\widetilde {s}_i\in U_{\alpha _i}U_{-\alpha _i}U_{\alpha _i}$ from [Reference MarquisMar18, Equation $7.34$ on page $135$ ], and is such that $\widetilde {w} := \widetilde {s}_{i_1} \cdots \widetilde {s}_{i_r}$ is independent of the choice of reduced decomposition $w = s_{i_1} \cdots s_{i_r}$ for $w\in W$ . The conjugation action of $N=\langle \{\widetilde {s}_i \ | \ i\in I\}\cup T\rangle \leq G$ on root groups is then compatible with the W-action on roots:

(6.3) $$ \begin{align} \widetilde{w}U_{\alpha}\widetilde{w}^{-1}=U_{w\alpha}\quad\text{for all } w\in W \text{ and } \alpha\in \Delta^{\operatorname{\mathrm{re}}}\cup\Delta^{\operatorname{\mathrm{im+}}}. \end{align} $$

More precisely, N admits an adjoint action $\operatorname {\mathrm {Ad}}_k\colon \thinspace N\to \mathrm {Aut}(\mathfrak {g}_k):\widetilde {w}\mapsto w^*$ on $\mathfrak {g}_k:=\mathfrak {g}_{\mathbb {Z}}\otimes _{\mathbb {Z}}k$ , such that

(6.4) $$ \begin{align} w^*\mathfrak{g}_{\alpha k}=\mathfrak{g}_{w\alpha k}\quad\text{for all } w\in W \text{ and } \alpha\in \Delta, \end{align} $$

where $\mathfrak {g}_{\alpha k}:=\mathfrak {g}_{\alpha \mathbb {Z}}\otimes _{\mathbb {Z}}k$ . Moreover, if $\alpha \in \Delta ^+$ and $x\in \mathcal {B}_{\mathbb {Z}}\cap \mathfrak {g}_{\alpha \mathbb {Z}}$ , then for all $\lambda _x\in k$ we have

(6.5) $$ \begin{align} \widetilde{w}[\exp](\lambda_xx)\widetilde{w}^{-1}=[\exp](\lambda_xw^*x)\quad\text{for all } w\in W \end{align} $$

(note that if $\alpha \in \Delta ^+$ is such that $w\alpha \in \Delta ^-$ , then necessarily $\alpha \in \Delta ^{\operatorname {\mathrm {re+}}}$ and the notation $[\exp ](\lambda _xw^*x)\in U_{w\alpha }$ still makes sense for $x\in \mathfrak {g}_{\alpha \mathbb {Z}}$ , as in this case $k\to U_{w\alpha }:\lambda \mapsto [\exp ](\lambda w^*x)$ is just a parametrization of $U_{w\alpha }$ as in [Reference MarquisMar18, §7.4.3 equation (7.26)]).

Finally, note that if $K \subseteq I$ , then $U^{\operatorname {\mathrm {ma}}}_{\Delta ^+_K}$ is normalized by $\widetilde {w}$ for any $w\in W_K$ by (6.3). As the K-residue $R_K(c)$ of $\mathcal B$ is given by $R_K(c) = U^{\operatorname {\mathrm {ma}}}_{\Delta ^+(K)}W_K c$ , and as $U^{\operatorname {\mathrm {ma}}}_{\Delta ^+_K}$ is normalized by $U^{\operatorname {\mathrm {ma}}}_{\Delta ^+(K)}$ (see [Reference MarquisMar18, Lemma $8.58(4)$ ]), we deduce that

(6.6) $$ \begin{align} U^{\operatorname{\mathrm{ma}}}_{\Delta^+_K} \leq \mathrm{Fix}(R_K(c)). \end{align} $$

Example 6.6. For $w\in W$ , the sets $\Delta _w:=\{\alpha \in \Delta ^+ \ | \ w^{-1} \alpha \in \Delta ^-\}$ and $\Delta ^+\setminus \Delta _w$ are both closed. In particular, there is a unique decomposition $U^{\operatorname {\mathrm {ma+}}}=U^{\operatorname {\mathrm {ma}}}_{\Delta _w}\cdot U^{\operatorname {\mathrm {ma}}}_{\Delta ^+\setminus \Delta _w}$ by Lemma 6.3. In view of Remark 6.5 and the fact that $U_- \cap U^{\operatorname {\mathrm {ma+}}} = \{1\}$ (see [Reference MarquisMar18, Theorem 8.78 and Definition B.30 axiom (RT3)]), we then have

(6.7) $$ \begin{align} U^{\operatorname{\mathrm{ma}}}_{\Delta^+\setminus\Delta_w}=U^{\operatorname{\mathrm{ma+}}}\cap \widetilde{w} U^{\operatorname{\mathrm{ma+}}}\widetilde{w}^{-1}. \end{align} $$

6.2 Lower bound for the nub

We start by obtaining a lower bound for the nub of an element $\widetilde {w}$ with $w\in W$ straight, by strengthening the results of §5.3, taking this time into account imaginary root groups as well.

Since W is a finitely generated linear group, there exists by Selberg’s lemma a torsion-free finite index normal subgroup $W_0$ of W, which we fix in the sequel. For $w\in W$ we define the essential support of w as

$$\begin{align*}\operatorname{\mathrm{esupp}}(w) := \bigcap_{n\in \mathbb{N}} \mathrm{supp}(w^n) \subseteq S. \end{align*}$$

We recall that a subset $J \subseteq S$ is essential if it is nonempty and has no spherical component.