Relative cubulation of relative strict hyperbolization

TL;DR

This paper demonstrates that many relatively hyperbolic groups from relative strict hyperbolization act cocompactly on CAT(0) cubical complexes, leading to new insights into their residual finiteness, virtual specialness, and applications in manifold theory.

Contribution

It establishes conditions under which these groups are residually finite and virtually special, and explores applications to non-positively curved manifolds and non-triangulable aspherical manifolds.

Findings

Many relatively hyperbolic groups admit cocompact CAT(0) cubical actions.

Under certain conditions, these groups are residually finite.

Applications include constructing new non-positively curved manifolds and non-triangulable aspherical manifolds.

Abstract

We prove that many relatively hyperbolic groups obtained by relative strict hyperbolization admit a cocompact action on a CAT(0) cubical complex. Under suitable assumptions on the peripheral subgroups, these groups are residually finite and even virtually special. We include some applications to the theory of manifolds, such as the construction of new non-positively curved Riemannian manifolds with residually finite fundamental group, and the existence of non-triangulable aspherical manifolds with virtually special fundamental group.