A combination theorem for cubulation in small cancellation theory over free products

TL;DR

This paper proves that certain groups formed by quotients of free products of cubulable groups under small cancellation conditions are themselves cubulable, expanding the class of relatively hyperbolic groups known to admit cubulations.

Contribution

It introduces a new cubulation theorem for groups with small cancellation over free products, without geometric assumptions on peripheral subgroups.

Findings

Establishes cubulability for a broad class of relatively hyperbolic groups.

Constructs wallspace structures combining walls from free factors and universal covers.

First cubulation result for relatively hyperbolic groups without geometric constraints on peripherals.

Abstract

We prove that a group obtained as a quotient of the free product of finitely many cubulable groups by a finite set of relators satisfying the classical $C'(1/6)$--small cancellation condition is cubulable. This yields a new large class of relatively hyperbolic groups that can be cubulated, and constitutes the first instance of a cubulability theorem for relatively hyperbolic groups which does not require any geometric assumption on the peripheral subgroups besides their cubulability. We do this by constructing appropriate wallspace structures for such groups, by combining walls of the free factors with walls coming from the universal cover of an associated $2$-complex of groups.