A super-rigid non-lattice
by
Hyman Bass
Columbia University

Call a group L linear if it has a faithful (finite dimensional) complex representation, and representation rigid if, in each dimension, there are only finitely many classes of simple representations. Platonov conjectured that if L is linear and representation rigid then L is essentially an arithmetic group. Lubotzky and I have constructed a counter-example. If F is the rank 1 form of the exceptional group F₄, and if H is a suitable cocompact lattice lattice in F, then we produce an L of infinite index in H ×H, and containing the diagonal, such that all representations of L extend uniquely to H ×H. This construction uses the fact that H is a hyperbolic group, a new theorem of Ol'shanskii about quotients of hyperbolic groups, and a theorem of Grothendieck connecting representation theory with profinite completions. The representation rigidity of L then follows from the super-rigidity theorem of Corlette and Gromov-Schoen, which says that H is super rigid in F.

Date received: May 7, 1998

Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. Document # cabb-46.