Decomposing CAT(0) Cube Complexes - O'Donnell, Christopher. (2017)
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 on 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.