Algebra Seminar

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 $V^W = {0}$. 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 V_C, the complexification of V, and m_C the complexification of m for m in M. Let Y_W = V_C - \cup_{M\in M}M_C and X_W = Y_W/W.

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