Visual boundaries of Diestel-Leader graphs

Keith Jones; Gregory A. Kelsey

Visual boundaries of Diestel-Leader graphs

Group Theory 2015-05-29 v2 Geometric Topology

Authors: Keith Jones , Gregory A. Kelsey

Abstract

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.

Cite

@article{arxiv.1307.2163,
  title  = {Visual boundaries of Diestel-Leader graphs},
  author = {Keith Jones and Gregory A. Kelsey},
  journal= {arXiv preprint arXiv:1307.2163},
  year   = {2015}
}

Comments

19 pages, 5 figures The substantive changes made in this revision are mostly found in Section 5.1, where we prove that geodesics in $\dl_d(q)$ have at most one "turn"

Visual boundaries of Diestel-Leader graphs

Abstract

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