Relative Cubulation of Small Cancellation Free Products

TL;DR

This paper demonstrates that small cancellation free products of groups admit relatively geometric actions on CAT(0) cube complexes, expanding the class of such groups and showing residual finiteness for certain free products.

Contribution

It proves that $C'(rac16)$--small cancellation free products are relatively cubulated, extending previous results and introducing a boundary separation approach for relative cubulation.

Findings

Relatively geometric actions are preserved under $C'(rac16)$--small cancellation free products.

Residually finite groups remain residually finite after small cancellation free products.

The boundary separation criterion effectively establishes relative cubulation in this setting.

Abstract

We expand the class of groups with relatively geometric actions on CAT(0) cube complexes by proving that it is closed under $C'(\frac16)$--small cancellation free products. We build upon a result of Martin and Steenbock who prove an analogous result in the more specialized setting of groups acting properly and cocompactly on CAT(0) cube complexes. Our methods make use of the same blown-up complex of groups to construct a candidate collection of walls. However, rather than arguing geometrically, we show relative cubulation by appealing to a boundary separation criterion and proving that wall stabilizers form a sufficiently rich family of full relatively quasiconvex codimension-one subgroups. The additional flexibility of relatively geometric actions has surprising new applications. In particular, we prove that $C'(\frac16)$--small cancellation free products of residually finite groups are residually finite.