Isometry groups of CAT(0) cube complexes

TL;DR

This paper investigates the relationship between automorphism groups and isometry groups of CAT(0) cube complexes, revealing conditions under which they coincide and exploring implications for hyperbolic and lattice structures.

Contribution

It establishes that non-coincidence of Aut(X) and Isom(X) implies a product decomposition with Euclidean space, and extends rank-rigidity results to lattices in isometry groups.

Findings

Aut(X) ≠ Isom(X) implies existence of a Euclidean factor in a subcomplex

Aut(X) = Isom(X) for hyperbolic, cocompact, 1-ended complexes unless quasi-isometric to hyperbolic plane

Rank-rigidity extends to lattices in Isom(X)

Abstract

Given a CAT(0) cube complex X, we show that if Aut(X) $\neq$ Isom(X) then there exists a full subcomplex of X which decomposes as a product with $\mathbb{R}^n$. As applications, we prove that if X is $\delta$-hyperbolic, cocompact and 1-ended, then Aut(X) $=$ Isom(X) unless X is quasi-isometric to $\mathbb{H}^2$, and extend the rank-rigidity result of Caprace-Sageev to any lattice $ \Gamma\leq$ Isom(X).