On sublinear bilipschitz equivalence of groups

TL;DR

This paper introduces sublinear bilipschitz equivalences (SBE) as a generalization of quasi-isometries, explores their implications for hyperbolic and nilpotent groups, and establishes invariants and properties related to large-scale geometry.

Contribution

It defines and studies SBEs, proves their effects on hyperbolic group boundaries, introduces a computable invariant for nilpotent groups, and explores large-scale geometric properties.

Findings

SBEs induce H"older homeomorphisms between hyperbolic group boundaries.

Subexponential growth is invariant under SBEs.

Nilpotent groups are quantitatively classified via a computable invariant $e_G$.

Abstract

We discuss the notion of sublinearly bilipschitz equivalences (SBE), which generalize quasi-isometries, allowing some additional terms that behave sublinearly with respect to the distance from the origin. Such maps were originally motivated by the fact they induce bilipschitz homeomorphisms between asymptotic cones. We prove here that for hyperbolic groups, they also induce H\"older homeomorphisms between the boundaries. This yields many basic examples of hyperbolic groups that are pairwise non-SBE. Besides, we check that subexponential growth is an SBE-invariant. The central part of the paper addresses nilpotent groups. While classification up to sublinearly bilipschitz equivalence is known in this case as a consequence of Pansu's theorems, its quantitative version is not. We introduce a computable algebraic invariant $e=e_G<1$ for every such group $G$, and check that $G$ is $O(r^e)$-bilipschitz equivalent to its associated Carnot group. Here $r\mapsto r^e$ is a quantitive sublinear bound. Finally, we define the notion of large-scale contractable and large-scale homothetic metric spaces. We check that these notions imply polynomial growth under general hypotheses, and formulate conjectures about groups with these properties.