3-manifolds everywhere

TL;DR

This paper demonstrates that random groups in certain models contain numerous subgroups isomorphic to fundamental groups of hyperbolic 3-manifolds with geodesic boundaries, embedded quasi-isometrically.

Contribution

It proves the existence of such subgroups in random groups across different models, extending understanding of their geometric subgroup structure.

Findings

Subgroups are isomorphic to hyperbolic 3-manifold groups

Subgroups are quasi-isometrically embedded

Results hold in both few relators and density models

Abstract

A random group contains many subgroups which are isomorphic to the fundamental group of a compact hyperbolic 3-manifold with totally geodesic boundary. These subgroups can be taken to be quasi-isometrically embedded. This is true both in the few relators model, and the density model of random groups (at any density less than a half).