Finite covers of graphs and cube complexes

Thesis

In this thesis we study finite covers of graphs, cube complexes and related spaces, and explore applications to rigidity of groups.

Leighton's Theorem states that two finite graphs with a common universal cover must have a common finite cover. We prove three generalisations of this. The first restricts how balls of a given size in the universal cover can map down to the two finite graphs when factoring through the common finite cover. The second generalises to covers of graphs of spaces that restrict to isomorphisms between vertex spaces. And thirdly, if the two finite graphs admit regular coverings by the same quasitree, then we arrange for the common finite cover to also be covered by this quasitree. In addition, we provide counter-examples to show that the assumptions in these generalisations cannot be relaxed.

Central to Haglund and Wise's theory of special cube complexes is the construction of the canonical completion and retraction, which enables one to build finite covers of special cube complexes in a highly controlled manner. We give a new approach to this construction with the idea of imitator covers. This idea provides greater insight into the construction and leads to powerful generalisations. It enables us to prove various results about finite covers of special cube complexes - most of which generalise existing theorems of Haglund and Wise to the non-hyperbolic setting. In particular, we prove a convex version of omnipotence for virtually special cubulated groups. Continuing the theme of special cube complexes, we give a novel account of Agol's proof that hyperbolic cubulated groups are virtually special; we retain the underlying ideas and constructions of Agol, but substantially change or add to many parts of the argument to give a more transparent and detailed account.

Finally, we apply Leighton-type arguments similar to those in the aforementioned results to obtain a quasi-isometric rigidity theorem. To be more precise, let G be a group that is one-ended, hyperbolic relative to virtually abelian subgroups, and has JSJ decomposition over two-ended subgroups containing only virtually free vertex groups that aren't quadratically hanging; we prove that any group quasi-isometric to G is abstractly commensurable to G. In particular, this theorem applies to certain "generic" HNN extensions of a free group over cyclic subgroups.

Authors: Shepherd, S

• DOI: 10.5287/ora-1z7gbzkvr
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: cube complexes; Leighton's Theorem; covering maps; quasi-isometric rigidity
• Subjects: Mathematics