Spectral computations on lamplighter groups and Diestel-Leader graphs

Metric Properties of Diestel-Leader Groups

In this paper we investigate metric properties of the groups $Γ_d(q)$ whose Cayley graphs are the Diestel-Leader graphs $DL_d(q)$ with respect to a given generating set $S_{d,q}$. These groups provide a geometric generalization of the family of lamplighter groups, whose Cayley graphs with respect to a certain generating set are the Diestel-Leader graphs $DL_2(q)$. Bartholdi, Neuhauser and Woess in \cite{BNW} show that for $d \geq 3$, $Γ_d(q)$ is of type $F_{d-1}$ but not $F_d$. We show below that these groups have dead end elements of arbitrary depth with respect to the generating set $S_{d,q}…

Visual boundaries of Diestel-Leader graphs

Diestel-Leader graphs are neither hyperbolic nor CAT(0), so their visual boundaries may be pathological. Indeed, we show that for $d>2$, $\partial\text{DL}_d(q)$ carries the indiscrete topology. On the other hand, $\partial\text{DL}_2(q)$, while not Hausdorff, is $T_1$, totally disconnected, and compact. Since $\text{DL}_2(q)$ is a Cayley graph of the lamplighter group $L_q$, we also obtain a nice description of $\partial\text{DL}_2(q)$ in terms of the lamp stand model of $L_q$ and discuss the dynamics of the action.

Automorphisms of Higher Rank Lamplighter Groups

Let $Γ_d(q)$ denote the group whose Cayley graph with respect to a particular generating set is the Diestel-Leader graph $DL_d(q)$, as described by Bartholdi, Neuhauser and Woess. We compute both $Aut(Γ_d(q))$ and $Out(Γ_d(q))$ for $d \geq 2$, and apply our results to count twisted conjugacy classes in these groups when $d \geq 3$. Specifically, we show that when $d \geq 3$, the groups $Γ_d(q)$ have property $R_{\infty}$, that is, every automorphism has an infinite number of twisted conjugacy classes. In contrast, when $d=2$ the lamplighter groups $Γ_2(q)=L_q = {\mathbb Z}_q \wr {\mathbb Z}$ h…

What is a horocyclic product, and how is it related to lamplighters?

This is a rather personal introductory outline of an interesting class of geometric, resp. graph- and group-theoretical structures. After an introductive section about their genesis, the general construction of horocyclic products is presented. Three closely related basic structures of this type are explained in more detail: Diestel-Leader graphs, treebolic spaces, and Sol-groups, resp. -manifolds. Emphasis is on their geometry, isometry groups, quasi-isometry classification and boundary at infinity. Subsequently, it is clarified under which parametrisation they admit discrete groups of isomet…

Simulations and the Lamplighter group

We introduce a notion of "simulation" for labelled graphs, in which edges of the simulated graph are realized by regular expressions in the simulating graph, and prove that the tiling problem (aka "domino problem") for the simulating graph is at least as difficult as that for the simulated graph. We apply this to the Cayley graph of the "lamplighter group" $L=\mathbb Z/2\wr\mathbb Z$, and more generally to "Diestel-Leader graphs". We prove that these graphs simulate the plane, and thus deduce that the seeded tiling problem is unsolvable on the group $L$. We note that $L$ does not contain any p…

Semidualities from products of trees

Let $K$ be a global function field of characteristic $p$, and let $Γ$ be a finite-index subgroup of an arithmetic group defined with respect to $K$ and such that any torsion element of $Γ$ is a $p$-torsion element. We define semiduality groups, and we show that $Γ$ is a $\mathbb{Z}[1/p]$-semiduality group if $Γ$ acts as a lattice on a product of trees. We also give other examples of semiduality groups, including lamplighter groups, Diestel-Leader groups, and countable sums of finite groups.

关于有限生成群中的线性发散性

On linear divergence in finitely generated groups

本文研究了有限生成群中线性发散性的性质，证明了 wreath 积、halo 积以及 Diestel-Leader 图等结构具有线性发散性，并推广了相关论证方法。

Coarse differentiation of quasi-isometries I: spaces not quasi-isometric to Cayley graphs

In this paper, we prove that certain spaces are not quasi-isometric to Cayley graphs of finitely generated groups. In particular, we answer a question of Woess and prove a conjecture of Diestel and Leader by showing that certain homogeneous graphs are not quasi-isometric to a Cayley graph of a finitely generated group. This paper is the first in a sequence of papers proving results announced in [EFW0]. In particular, this paper contains many steps in the proofs of quasi-isometric rigidity of lattices in Sol and of the quasi-isometry classification of lamplighter groups. The proofs of those res…