Quotient-saturated groups

Abstract

We introduce the new notion of quotient-saturation as a measure of the immensity of the quotient structure of a group, and we prove that it is a necessary condition for a non-elementary finitely presented group to embed in a hyperbolic group. More generally, we present a sufficient condition — called Congruence Extension Property equipment (in short, CEP-equipment) — for a finitely presented group to be quotient-saturated. Using this property, we deduce that non-elementary finitely presented subgroups of a hyperbolic group (in particular, non-elementary hyperbolic groups themselves) are quotient-saturated. Finally, we elaborate on the previous results to extend the scope of CEP-equipment (and hence of quotient-saturation) to finitely presented acylindrically hyperbolic groups.

The first named author thanks the support from the Universitat Politècnica de Catalunya (UPC) through a María Zambrano grant. The second named author thanks the support from UPC through a Margarita Salas grant. The three authors acknowledge support from the Spanish Agencia Estatal de Investigación through grant PID2021-126851NB-100 (AEI/ FEDER, UE).

Peer Reviewed

Postprint (published version)