Decomposing CAT(0) Cube Complexes
Abstract: By work of Caprace and Sageev ([CS11]), every finite-dimensional
CAT(0) cube complex X has a canonical product decomposition X = X 1 ×· · ·×X n into
irreducible factors and any group G which acts on X must have a finite index subgroup which
embeds in Aut(X 1 ) × · · · × Aut(X n ). We explore the natural followup questions, "If G =
G 1 × · · · × G n acts geometrically and essentially ... read moreon a CAT(0) cube complex X, does X
have product decomposition X = X 1 × · · · × X n ? If so, how close is the action to a
product action?" In chapter 3, we answer this question when each G i is a non-elementary
hyper- bolic group. We recover the canonical product decomposition X = X 1 × · · · × X n
and show that G has a finite index subgroup which acts with the product action on this
decomposition of X. In chapter 4, we create a generalization of the work in chapter 3 to
the case when each G i satisfies a property we call (AIP). Essentially, this property gives
us a large degree of control over how locally maximal abelian subgroups can intersect. We
again show that X has a decomposition X = X 1 × · · · × X n , though this is not the
canonical decomposition. We also show that G has a finite index subgroup which acts with
the product action on X. In chapter 5, we consider groups of the form Γ = G × A, where G
satisfies (AIP) and A is free-abelian. We show that any cube complex X on which Γ acts
geometrically and essentially decomposes as X = XG × XA , and there is a finite index
subgroup G0 × A0 of Γ which acts with a product action on XG × XA .
Thesis (Ph.D.)--Tufts University, 2017.
Submitted to the Dept. of Mathematics.
Advisor: Genevieve Walsh.
Committee: Robert Kropholler, Kim Ruane, and Talia Fernos.
Keyword: Mathematics.read less