Sydney University Algebra Seminar

# Algebra Seminar

### Buildings associated with Artin-Tits groups of spherical type.

Friday 5th April, 12-1pm, Carslaw 373.

Let V be a finite dimensional vector space over R and let W be a finite subgroup of GL(V) generated by reflections such that $VW = {0}$. We can associate to V a Euclidean structure which is W invariant. Denote by M the set of hyperplanes associated to the orthogonal reflections of W. Consider VC, the complexification of V, and mC the complexification of m for m in M. Let YW = V_C - \cup_{M\in M}MC and XW = YW/W.

Deligne (Invent. 1972) proved that XW is a K(\pi,1)-space and that \Pi1(XW, x) is the Artin-Tits group associated to W for any x\in XW. We will describe the notion of "gallery" he introduced in its proof and explain the main ideas of its proof.