Poincar\'e profiles of lamplighter diagonal products

TL;DR

This paper constructs finitely generated groups with specific Poincaré profiles, demonstrating sharp bounds and applications in geometric group theory, including non-embeddability results and graph distortion properties.

Contribution

It provides explicit constructions of groups with prescribed Poincaré profiles, extending the understanding of their possible behaviors and applications.

Findings

Existence of groups with Poincaré profiles between n/ log n and linear.

Sharpness of profiles for functions at least n/ (log log n).

Examples of graphs with large distortion in L^p spaces.

Abstract

We exhibit finitely generated groups with prescribed Poincar\'e profiles. It can be prescribed for functions between $n/\log n$ and linear, and is sharp for functions at least $n/(\log\log n)$. Those profiles were introduced by Hume, Mackay and Tessera in 2019 as a generalization of the separation profile, defined by Benjamini, Schramm and Tim\'ar in 2012. The family of groups used is based on a construction of Brieussel and Zheng. As applications, we show that there exists bounded degrees graphs of asymptotic dimension one that do not coarsely embed in any finite product of bounded degrees trees, and exhibit hyperfinite sequences of graphs of arbitrary large distortion in $L^p$-spaces.