On equivariant embeddings of generalized Baumslag-Solitar groups

TL;DR

This paper characterizes when certain generalized Baumslag-Solitar groups have the Haagerup Property, linking it to weak amenability, and explores their geometric embeddings and compression rates.

Contribution

It provides a complete characterization of the Haagerup Property for these groups and establishes new results on their L^p-compression and quasi-isometric embeddings.

Findings

G has the Haagerup Property iff G is weakly amenable.

G has the Haagerup Property if d=0, d=1, or n=1.

The L^p-compression rate of G is 1; equivariant rate is max{1/p,1/2} for non-amenable G.

Abstract

Let G be a group acting cocompactly without inversion on a tree X, with all vertex and edge stabilizers isomorphic to the same free abelian group Z^n. We prove that G has the Haagerup Property if and only if G is weakly amenable, and we give a necessary and sufficient condition for this to happen. In particular, denoting by d the rank of the fundamental group of the graph X modded out by G, we deduce that G has the Haagerup Property if either d=0, d=1, or n=1. In these three cases, we show that the L^p-compression rate of G is 1, and that its equivariant L^p-compression rate is max{1/p,1/2} (provided G is non-amenable). We also discuss quasi-isometric embeddings of G into a product of finitely many regular trivalent trees.