Ph.D Thesis

Ph.D StudentLazarovich Nir
SubjectNon-Positively Curved Regular Complexes
DepartmentDepartment of Mathematics
Supervisor PROF. Michah Sageev


The main goal of this thesis is to study regular CAT(0) complexes. These are complexes of non-positive curvature that locally look the same at each vertex, i.e, they have the same link at each vertex and the same polyhedral cells. We restrict our attention to two cases in which a combinatorial condition for local CAT(0) is known to exist: polygonal and cubical complexes.

Unlike their one-dimensional analogues, regular trees, specifying the isomorphism class of the link does not necessarily uniquely determine the complex. Thus it is natural to ask when does the local data determine the global structure of the complex, and when not, how many complexes with that local data are there. To date, there have been a few partial results pertaining to this question.

In this thesis we provide a necessary and sufficient combinatorial condition on the link, for the existence of a unique regular CAT(0) cube complex and a unique regular even-sided-polygonal CAT(0) complex. For an odd-sided-polygonal complex we show that our condition is sufficient for uniqueness. We also prove that when this condition is not met, there is a continuum of non-isomorphic regular even-sided-polygonal CAT(0) complexes.

We provide examples of links satisfying our condition and analyze the automorphism groups of the associated unique regular CAT(0) cube complexes. Generalizing the simplicity theorem of Tits and of Haglund-Paulin, we show that the automorphism group of these complexes is often virtually simple.