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

Connor MacMahon

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arXiv · 2605.20397

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

Connor MacMahon

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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.

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Connor MacMahon. 2026-05-19. An $\ell^2$ Obstruction for Elementary Embeddings of Hyperbolic Groups. https://arxiv.org/abs/2605.20397

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