Buildings, surfaces and quaternions

Alina Vdovina (CUNY)

Abstract: Buildings are very rich objects possessing geometric, algebraic, analytic and number theoretical aspects. We present explicit constructions of new infinite families of CW-complexes of arbitrary dimension with buildings as the universal covers. As an illustration of an analytic side of buildings we will review the superrigidity results of Gromov-Schoen and Daskalopoulos-Mese-Vdovina. For the arithmetic side of buildings, we will present the first infinite series of quaternionic groups of finite characteristic and non-residually finite groups acting on cube complexes, joint results with Rungtanapirom and Stix. Quotients of these groups give rise to new series of examples of Ramanujan complexes. We plan to finish with a long list of open problems which can be tackled by our methods.