From a question of Serre to the Quillen-Wendt conjecture
Alexander D. Rahm (National University of Ireland at Galway )
Wednesday 29th April, 2015 16:00-17:00 Maths 516
The (co)homology of the Bianchi groups has been the subject to a question by Serre, which was open for 40 years, namely on specifying the kernel of the map induced on homology by attaching the Borel-Serre boundary to the symmetric space quotient of the Bianchi groups.
This question been given a constructive answer by the speaker. Moreover, the studies of the latter on the (co)homology of the Bianchi groups have given rise to a new technique (called Torsion Subcomplex Reduction) for computing the Farrell-Tate cohomology of discrete groups acting on suitable cell complexes. This technique has not only already yielded general formulae for the cohomology of the tetrahedral Coxeter groups as well as, above the virtual cohomological dimension, of the Bianchi groups (and at odd torsion, more generally of SL_2 groups over arbitrary number fields), it also very recently has allowed Wendt to reach a new perspective on the Quillen conjecture; gaining structural insights and finding a variant that can take account of all known types of counterexamples to the Quillen conjecture. If no counterexample of completely new type surprisingly shows up, then this refined conjecture must be valid.