Premier exposé de 10h à 10h40.
Second exposé de 10h50 à 11h30.

Sunghwan Ko

Title : Finiteness of integral representations coming from Coxeter truncation polytopes

Abstract : Let P be a Coxeter polytope and S be the set of its facets. Let (\sigma_s)_{s\in S} be reflections in the facets. By Vinberg’s work, we can construct a discrete, faithful representation \rho : W_S \rightarrow \Gamma_S \in SL(V) where W_S is an associated Coxeter group of P, \Gamma_S is a group generated by \sigma_s and V is a real vector space of dimension |S|+1. This construction gives rise to the representation variety of the Coxeter polytope P.

In this talk, I will focus on Coxeter truncation polytopes, which are obtained from a loxodromic Coxeter simplex by successively truncating vertices. I will introduce the notion of Cartan matrix as a key tool of investigating Vinberg representations, and briefly sketch the proof that there are only finitely many inequivalent integral representations coming from a Coxeter truncation polytope of dimension 3.

Seunghoon Hwang

Title: Projective reflection groups with finite-covolumes

Abstract: In 1971, Vinberg introduced “projective reflection groups”, which are realizations of abstract Coxeter groups as discrete reflection groups of “projective Coxeter polytopes”. Among these, “loxodromic” ones produce convex projective orbifolds and a natural way of measuring their “covolumes”. Furthermore, in 2017, Marquis proved that a loxodromic, “2-perfect” projective reflection group has finite-covolume if and only if it is “quasi-perfect”. In this talk, I will explain this result along with the related definitions and ideas, and introduce a question that arises naturally from it.