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

The first ${\mathrm{\ell }}^2$ 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.