Finite Stature in Graphs of Cube Complexes with Cyclonormal Edges

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 $X$.