Groups and buildings
Final Report Abstract
Uniform lattices on exotic buildings form an exciting class of groups because they are on one hand closely related to arithmetic groups while on the other hand conjecturally virtually simple. The buildings they are acting on provide explicit geometric models which are not available for (other) finitely presented simple groups. We demonstrate the usefulness of these geometric models by providing detailed information for a particular class of uniform building lattices, the Singer cyclic lattices. We show that their number grows super-exponentially with the parameter q. Since, by work of Radu, the number of arithmetic lattices grows only polynomially with q, this implies that the number of exotic Singer cyclic lattices grows super-exponentially as well, even up to quasi-isometry. For q ≤ 5 we provide precise information about every individual Singer cyclic lattice. In particular, we show that no two of these are quasi-isometric to each other and that for each q exactly one is arithmetic while all others are exotic. The main problem that remains open is to prove that exotic building lattices are virtually simple, or even to prove it in a single instance. The difficulty lies in finding an obstruction to residual finiteness that sees the difference between arithmetic and non-arithmetic lattices.
Publications
- On panel-regular A2 lattices, Geom. Dedicata 191 (2017), 85–135
Stefan Witzel
(See online at https://doi.org/10.1007/s10711-017-0247-8)
