Pattern preserving quasi-isometries in lamplighter groups and other related groups

Abstract

In this paper we explore the interplay between aspects of the geometry and algebra of three families of groups of the form B semidirect the integers Z, namely Lamplighter groups, solvable Baumslag-Solitar groups and lattices in SOL. In particular we examine what kind of maps are induced on B by quasi-isometries that coarsely permute cosets of the Z subgroup. By the results of Schwartz(1996) and Taback(2000) in the lattice in SOL and solvable Baumslag-Solitar cases respectively such quasi-isometries induce affine maps of B. We show that this is no longer true in the lamplighter case but the induced maps do share some features with affine maps.

Figures (6)

• Figure 1: Top: The Diestel-Leader graph as a horocyclic product two trees. Middle: The model space for $BS(1,2)$ as a horocyclic product of the hyperbolic plane with a tree. Bottom: The model space of a lattice in SOL as a horocyclic product of two hyperbolic planes. Note a part of this figure is taken from a figure in Farb1998
• Figure 2: The distance between geodesics $p$ and $q$ is given by the length of the segment from $B$ to $C$ (or equivalently $A$ to $D$). If $h$ denotes the height function on $DL(n,n)$. Then $d_{\ell}(p,q)=n^{h(B)}$ and $d_u(p, q)=n^{- h(C)}$ so that $\delta(p,q)=n^{h(B)-h(C)}$ and therefore $\log_n(\delta(p,q))$ is exactly the distance between the geodesics $p$ and $q$.
• Figure 3: A schematic of an $(\epsilon,M)$ quadrilateral in the model space for $BS(1,2)$, given by vertical geodesics $0,v,w,$ and $v+w$. Every dashed grey segment has length 1. The coarse height $t_v$ is the height where the geodesics 0 and $v$ are closest together. At $t_v$, we can see that the distance between $0$ and $v$ is small and the distance between $w$ and $v+w$ is small but the distance between $0$ and $v+w$ is large. The distance between $0$ and $w$ is realized at height $t_w$, which is also a small distance.
• Figure 4: The shaded region, $A$, bounded by the black hyperbolas centered around the origin consists of points $p$ with $\delta(0,p)<\epsilon'$. While regions $B$(red and dashed) and $C$ (blue and cross-hatched) are those points $p$ with $\delta(b,p)<\epsilon'$ and $\delta(c,p)<\epsilon'$ respectively. If $M$ is large enough then the intersection of $B$ and $C$ has two regions, and each of these regions contains only one lattice point.
• Figure 5: A height one portion of $DL(2,2)$ on the right and its tree factors on the left. Vertex labels illustrate the proof of Theorem \ref{['thrm:affineisometry']}. The red (dotted) line represents the vertical geodesic corresponding to the identity coset in the right-hand factor.

Theorems & Definitions (31)

• Theorem 1.1
• Theorem 1.2
• Lemma 1.3
• Definition 2.1: Quasi-isometry
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Lemma 2.6
• proof