Fixed points of finite subgroups in the infinite dihedral tree and their cell stabilizers

Clause (i). By [F1] every element has a unique reduced syllable word. A word of length $1$ is a generator and is an involution. If an odd reduced word $w=g_1\cdots g_k$ has $k\ge 3$, then $g_k=g_1$ and $g_{1}w{g}_1=g_2\cdots g_{k-1}$ is a shorter odd reduced word. Induction shows that $w$ is conjugate to $s$ or $t$, so has order $2$. If $k=0$, $w=1$. If $k=2j>0$, the word is $(st{)}^j$ or $(ts{)}^j=t(st{)}^{j}t$; [F1] says $st$ has infinite order, so every such word has infinite order. Thus every element is the identity, a reflection, or an infinite-order translation. If ${\rho }_1\mathrm{\ne }{\rho }_2$ are reflections, then ${\rho }_1{\rho }_2\mathrm{\ne }1$ and its reduced syllable word is obtained from a concatenation of two odd-length words by cancelling equal adjacent pairs. The resulting word is nonempty and has positive even length, so it is a translation of infinite order by the preceding classification. A finite subgroup therefore contains no translation and at most one reflection, so it is trivial or has order two generated by a reflection.

The line structure. By [F2] the only spherical subsets are $\mathrm{\varnothing },\{s\},\{t\}$, so there are only vertices and edges, and by [F3] the $1$-skeleton is $G=\mathrm{Cay}(W,\{s,t\})$. Let $p:=st$ and, for $n\in \mathbb{Z}$, set $v_{2n}:=p^n$ and $v_{2n+1}:=p^{n}s$. The normal forms of [F1] show these vertices are distinct and exhaust $W$: even reduced words are $p^n$; odd words starting with $s$ are $p^{n}s$ for $n\ge 0$, and odd words starting with $t$ are $p^{n}s=(ts{)}^{-n-1}t$ for $n<0$. Consecutive vertices differ by right multiplication by $s$ and $t$, respectively, so this indexes the Cayley graph as a bi-infinite line. Send $v_k$ to $k$ and extend linearly over each edge. Each edge has length $1$ by [F3]. For any cell chain, the sum of its cellwise lengths is at least the absolute difference of the endpoint coordinates by the triangle inequality on $\mathbb{R}$; conversely, the finite line segment between two points is a finite chain of edge subsegments with length equal to that coordinate difference. Thus [F8] gives the chain metric as the usual metric on $\mathbb{R}$, and $\mathrm{\Sigma }$ is a metric line. Every isometry of this line is either a translation or a reflection: after writing $f(0)=c$, one has $f(1)=c+1$ or $c-1$, and the distances to these two points determine $f(x)$ uniquely as $c+x$ or $c-x$. Therefore a nonidentity involution is a reflection with exactly one fixed point. The action on vertices is free by [F4].

Clause (ii). Let $w\in W$. By [F2] the vertex $w$ is the $0$-cell $w{W}_{\mathrm{\varnothing }}$, whose relative interior is $\{w\}$; by [F4] applied with $q=w{W}_{\mathrm{\varnothing }}$ and $T=\mathrm{\varnothing }$, ${\mathrm{Stab}}_W(w)=\{1\}$. A finite subgroup fixing a vertex $v$ lies in ${\mathrm{Stab}}_W(v)=\{1\}$, so it is trivial.

Clause (iv). Assume now that $(W,S)$ is finite with $S=\{s,t\}$ and $m(s,t)=3$. Then $B$ is positive definite by [F6], $S$ is spherical and $W=W_S$, so every spherical coset is contained in $W_S=W$ and indexes a face of the single top cell $C_S$ by The cellulation of the Davis complex: incidence, stabilizers and the model U(W,K) (1),(2). Hence $\mathrm{\Sigma }=C_S$; the universal-system description in [F2] is not used in this finite case. The point $0$ lies in the interior of $C_S$ by [F6], so in the top-cell chart it has $w_0=1$ and $I=S$; the point-stabilizer formula of [F4] gives ${\mathrm{Stab}}_W(0)=W_S=W$. Here $W$ is dihedral of order $2m=6$ by [F6], and $W=W_S$ is a spherical parabolic of full rank.

Clause (iii). Let $w$ be a nonidentity reflection. By step 1.1 it is an involution; by step 1.2 its action on the line is a reflection with a unique fixed point $m$. By step 1.3 no nontrivial element fixes a vertex, so $m$ lies in the interior of an edge. The edge enumeration in step 1.2 writes its carrier as $q=p^k{W}_r$ for some $k\in \mathbb{Z}$, $r\in \{s,t\}$, with endpoints $a=p^k$ and $b=p^{k}r$. The reflection swaps these endpoints, so $wa=b$ and $w=b{a}^{-1}=p^{k}r{p}^{-k}$. In the $\dot{q}$-chart the midpoint has coordinate $0\in C_{\{r\}}$, in the chamber face $w_0=1$, $I=\{r\}$. The formula of [F4] gives ${\mathrm{Stab}}_W(m)=(\dot{q})W_r(\dot{q}{)}^{-1}$. Since $\dot{q}\in p^k{W}_r$, this is $p^k{W}_r{p}^{-k}=\{1,w\}=⟨w⟩$; it is the setwise stabilizer of the carrier edge and a conjugate of the rank-one spherical parabolic $W_r$ by [F5]. Under AC, this is the equality case of the general point-stabilizer formula [F7]; the explicit computation here does not rely on that theorem.

Clause (v) and the Choice bookkeeping. Steps 1.1-1.4 and 2.1 compute the two examples: the finite subgroups of the universal rank-two group are trivial or generated by one reflection, each nontrivial finite subgroup fixing the midpoint of a $1$-cell with stabilizer the conjugate spherical parabolic $p^k{W}_r{p}^{-k}$; in the finite rank-two case the whole group fixes the interior point $0$ of the top cell, with stabilizer $W_S=W$. These are the rank-one and top-rank extremes of the containment statement in [F7]. The normal-form, line, midpoint and cell calculations are choice-free. AC is assumed only for the comparison with [F7], and no conclusion of that general theorem is used in the explicit proof. No assertion is made about infinite subgroups, higher-rank trees or the number of conjugacy classes of finite subgroups in general.