An $\ell^2$ Obstruction for Elementary Embeddings of Hyperbolic Groups

TL;DR

This paper investigates how the first  Betti number behaves under elementary embeddings in hyperbolic groups, establishing strict inequalities and monotonicity properties in specific cases.

Contribution

It proves a strict inequality for elementary embeddings of non-abelian subgroups in torsion-free hyperbolic groups and demonstrates monotonicity for existential embeddings.

Findings

Strict inequality for elementary embeddings in hyperbolic groups

Monotonicity of  Betti number for existential embeddings

Classification-based proof using Perin classification

Abstract

The first $\ell^2$ Betti number of a group is non-decreasing under various embeddings arising from first order logic. Strict inequality is proved for elementary embeddings of non-abelian proper subgroups within torsion free hyperbolic groups using Perin's classification of such inclusions. The monotonicity is further demonstrated for existential embeddings of arbitrary finitely generated groups.