Speaker
Description
During the last fifteen years, computer-assisted proofs have lead to the proofs of property (T) for several classes of groups. This property is in turn related to expanders, that is infinite families of sparse graphs with growing number of vertices and good connectivity properties. The quality of connectivity properties can be further estimated by means of Kazhdan constants. The common approach in such computer-assisted proofs is a usage of semi-positive definite programming. One can characterize property (T) by positivity of specific matrices originating from the group rings associated to groups. Due to the works of N. Ozawa, and recently U. Bader and P. W. Nowak, one can distinguish two alternative conditions for property (T): the first one making use of generators of a group and the second one involving group presentation. In both approaches one counts for an explicit estimate for the Kazhdan constant. One achieves this by estimating the spectral gap of a specific Laplace operator, defined in the group ring setting. In this talk, I am going to prove the estimates for the spectral gaps of the Laplace operators coming from the presentation of the symplectic groups over integers. This is a natural continuation of my previous result on this topic concerning among others special linear groups. We customize the induction technique and reduce the proof to specific computations for particular low-degree symplectic groups.
Another important aspect of this approach is to turn numerical solutions for such specific computations into a rigorous mathematical proof. This is done by the certification strategy (applied before by e.g. M. Kaluba, D. Kielak, P. W. Nowak, N. Ozawa). We comment here on our recent improvement of this strategy which can lead to faster computations. In order to accelerate them, we also applied Wedderburn decomposition.
If time permits, I would like to discuss possibilities of modifying semi-positive definite problems in a way that they would provide us with human-readible solutions. In this approach I would propose adding some type of discrete objective function to the corresponding simplex-like problem. This is a joint work with Jakub Szymański.
