Random groups and nonarchimedean lattices

TL;DR

This paper introduces models of random groups of intermediate rank, extending Gromov's constructions, with a focus on higher rank lattices in algebraic groups over local fields.

Contribution

It presents a novel higher rank model of random groups based on lattices in algebraic groups over local fields, expanding the scope of random group theory.

Findings

Models include a density model for intermediate rank groups

Random groups are higher rank analogs of Gromov's constructions

Lattices in algebraic groups over local fields form the basis of these models

Abstract

We consider models of random groups in which the typical group is of intermediate rank (in particular, it is not hyperbolic). These models are parallel to M. Gromov's well-known constructions and include for example a "density model" for groups of intermediate rank. The main novelty is the higher rank nature of the random groups. They are randomization of certain families of lattices in algebraic groups (of rank 2) over local fields.