[Paper Review] Cross-wired lamplighter groups

Yves Cornulier, David Fisher+1 more|arXiv (Cornell University)|Jun 3, 2012

Electromagnetic Scattering and Analysis13 references3 citations

TL;DR

This paper introduces cross-wired lamplighter groups as a generalization of standard lamplighter groups $F\wr\mathbb{Z}$, characterizing closed, cocompact subgroups of the isometry group of Diestel-Leader graphs $\mathrm{DL}(m,n)$. It proves that every such group has an index-two subgroup acting transitively on another Diestel-Leader graph, showing that lattices in $\mathrm{Isom}(\mathrm{DL}(n,n))$ are not necessarily lamplighter groups, and providing an algebraic characterization via semidirect products with specific self-similar subgroups.

ABSTRACT

We give a necessary and sufficient condition for a locally compact group to be isomorphic to a closed cocompact subgroup in the isometry group of a Diestel-Leader graph. As a consequence of this condition, we see that every cocompact lattice in the isometry group of a Diestel-Leader graph admits a transitive, proper action on some other Diestel-Leader graph. We also give some examples of lattices that are not virtually lamplighters. This implies the class of discrete groups commensurable to lamplighter groups is not closed under quasi-isometries and, combined with work of Eskin, Fisher and Whyte, gives a characterization of their quasi-isometry class.

Motivation & Objective

• To characterize closed, cocompact subgroups of $\mathrm{Isom}^+(\mathrm{DL}(m,n))$ for $m,n \geq 2$.
• To determine when such subgroups arise as lattices and whether they are quasi-isometric to standard lamplighter groups.
• To provide an algebraic characterization of lattices in $\mathrm{Isom}(\mathrm{DL}(n,n))$ beyond the standard wreath product structure.
• To show that every cocompact lattice in $\mathrm{Isom}(\mathrm{DL}(n,n))$ admits a transitive, proper action on some other Diestel-Leader graph.
• To construct examples of lattices that are not virtually lamplighters, demonstrating that the class of groups commensurable to lamplighter groups is not closed under quasi-isometries.

Proposed method

• Use the structure of Diestel-Leader graphs $\mathrm{DL}(m,n)$ as level sets of Busemann functions on products of $n+1$-regular trees.
• Analyze isometries of $\mathrm{DL}(m,n)$, distinguishing between positive isometries (restrictions of product automorphisms) and non-positive ones (compositions with the flip).
• Establish a semidirect product decomposition $\Gamma = H \rtimes \langle t \rangle$ for closed, cocompact subgroups $\Gamma < \mathrm{Isom}^+(\mathrm{DL}(m,n))$, where $H$ is a normal, non-compact open subgroup.
• Define open subgroups $L, L' \subset H$ such that $tLt^{-1}$ and $t^{-1}L't$ are finite-index subgroups of index $m$ and $n$, respectively, and the unions $\bigcup_k t^{-k}Lt^k$ and $\bigcup_k t^kL't^{-k}$ exhaust $H$.
• Show that $L \cap L'$ is compact and $LL' = H$, implying $L \backslash H / L'$ is a single point.
• Construct a proper, transitive action of $\Gamma$ on $\mathrm{DL}(m,n)$ with kernel $\bigcap_k t^{-k}(L \cap L')t^k$, using the self-similar structure of $H$.

Experimental results

Research questions

• RQ1 What characterizes closed, cocompact subgroups of $\mathrm{Isom}^+(\mathrm{DL}(m,n))$ for $m,n \geq 2$?
• RQ2 Can every cocompact lattice in $\mathrm{Isom}(\mathrm{DL}(n,n))$ act transitively on another Diestel-Leader graph?
• RQ3 Are there lattices in $\mathrm{Isom}(\mathrm{DL}(n,n))$ that are not virtually isomorphic to standard lamplighter groups $F\wr\mathbb{Z}$?
• RQ4 Is the class of groups commensurable to lamplighter groups closed under quasi-isometries?
• RQ5 What is the algebraic structure of lattices in $\mathrm{Isom}(\mathrm{DL}(m,n))$ when $m \neq n$?

Key findings

• Every closed, cocompact subgroup $\Gamma < \mathrm{Isom}^+(\mathrm{DL}(m,n))$ has a unique open normal subgroup $H$ such that $\Gamma/H \cong \mathbb{Z}$, and $H$ decomposes as a union of conjugates of two subgroups $L$ and $L'$ with specific index conditions.
• The double coset space $L \backslash H / L'$ is a single point, and $L \cap L'$ is compact, indicating a highly symmetric, self-similar structure in $H$.
• Any cocompact lattice $\Gamma < \mathrm{Isom}(\mathrm{DL}(n,n))$ admits a proper, transitive action on another Diestel-Leader graph $\mathrm{DL}(n,n)$, even though $\Gamma$ may not be a lamplighter group.
• There exist lattices in $\mathrm{Isom}(\mathrm{DL}(n,n))$ that are not virtually lamplighter groups, showing that the class of groups commensurable to lamplighter groups is not closed under quasi-isometries.
• When $\Gamma$ is discrete, the parameters $m$ and $n$ must be equal, so $\Gamma < \mathrm{Isom}(\mathrm{DL}(n,n))$, and $\Gamma$ is finitely generated.
• Examples of cross-wired lamplighters include linear groups arising from Heisenberg groups over local fields of positive characteristic, which are residually finite, and non-linear examples can be constructed from nilpotent groups with contracting automorphisms.