Finite Stature in Graphs of Cube Complexes with Cyclonormal Edges

TL;DR

This paper proves that certain cube complexes with specific subgroup properties have finite stature, leading to their virtual specialness when vertex groups are hyperbolic, advancing understanding of their geometric group structure.

Contribution

It establishes finite stature for cube complexes with cyclonormal edge groups, enabling conclusions about their virtual specialness in hyperbolic cases.

Findings

Finite stature is proven for the fundamental group of the complex.

This result implies virtual specialness when vertex groups are hyperbolic.

The work connects subgroup properties to large-scale geometric features.

Abstract

Given a compact cube complex $X$ that splits as a graph of virtually special cube complexes. Suppose that the fundamental groups of edge spaces are cyclonormal in the fundamental groups of adjacent vertex spaces. We show that $\pi_1X$ has finite stature with respect to vertex groups in the sense of Huang-Wise. In particular, when vertex groups are hyperbolic, this allows us to deduce virtual specialness of $X$.