Ph.D Student | Izhar Oppenheim |
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Subject | Groups Acting on Simplicial Complexes |

Department | Department of Mathematics |

Supervisor | Professor Bader Uri |

Full Thesis text |

This thesis deals with criteria for property (T) and the vanishing of L^2 cohomologies for groups acting on simplicial complexes (the reader should note that the vanishing of the first L^2 cohomology is equivalent for property (T)).

The idea is that for a 'nice' group acting 'well' on a 'nice' simplicial complex there are some properties of the group which one might detect looking only at the local geometry of the simplicial complex and that property (T) and the vanishing of L^2 cohomologies are such properties. Key results of this nature (regarding property (T) and the vanishing of L^2 cohomology) where proven by Ballmann and Swiatkowski and independently Zuk (both relaying on the work of Garland) and by Kassabov.

In our work, we generalize the results mentioned above and give new criteria for property (T) and for the vanishing of cohomologies. The idea is to study geometrical properties of the links of vertices such as the Laplapcian eigenvalues or an 'angle' of those links to get a sufficient condition for property (T) and the vanishing of L^2 cohomologies. Our generalizations improve the previous criteria in several aspects:

1. By taking 'average' of Laplacian eigenvalues of links we give a new criterion for property (T), which proves property (T) and the vanishing of L^2 cohomologies for new interesting examples not covered by the previous criteria.

2. We give a vanishing criterion which implies the vanishing of all L^2 cohomologies and is more checkable than previous criteria.

3. We give a checkable criterion for property (T) that works for simplicial complexes which are not locally finite (as opposed to previous criteria by Ballmann and Swiatkowski).

4. We generalize the definition of an 'angle' of a link and state and prove a 'Kassabov type' criterion for this new definition .