# Simplicial complexes workshop

There will be one talk each day from 11:00 - 12:00, and we'll go for lunch together afterwards. If you'd like to join meet us in Aud 1 in August Krogh Building at that time (and don't forget to bring a cup of coffee!)

**Monday, August 17th 2020, 11:00 - 12:00**

**On the coset poset of 2-nilpotent subgroups in a 3-nilpotent group (Simon Gritschacher)**

To a group G and a family of subgroups F of G one can associate a simplicial complex C(F,G) constructed from the poset of all cosets gH where g \in G and H runs through F. It is a natural question to ask how the topological properties of C(F,G) are related to the algebraic properties of F and G, in particular with regards to higher connectivity of C(F,G). I will look at the particular case where G is a nilpotent group of class at most 3 and F the family of subgroups of class at most 2. For certain G, 1- and 2-connectivity of C(F,G) detect whether G is a 2-Engel or 2-nilpotent group, respectively. This addresses a question of Okay and is based on discussions with Bernardo Villarreal.

---------------------------------------------------------

**Tuesday, August 18th 2020, 11:00 - 12:00**

**Discrete Morse Theory and E^k buildings (Peter Patzt)**

**Wednesday, ****August 19th 2020, 11:00 - 12:00Coset complexes (Benjamin Brück)**

Coset complexes provide one with a possibility to construct for a given group G simplicial complexes that are equipped with an action of G and have a prescribed set of simplex stabilisers. These complexes have nice combinatorial and topological properties: They are completely balanced, facet transitive and whether they are connected or simply connected can easily be expressed in terms of the simplex stabilisers of the G-action. Examples of this construction include Coxeter complexes (G = a Coxeter group), Tits buildings (G = GL_n or more generally a group with BN pair) and the free factor complex (G = Aut(F_n) ).

The talk will be a short introduction to this family complexes. I will present some of their properties, give examples and in particular talk about the case of coset complexes that are Cohen-Macaulay.

---------------------------------------------------------

**Thursday, August 20th, 11:00 - 12:00Centrally symmetric complexes and the minimal triangulation of RP^d (Hailun Zheng)**

---------------------------------------------------------

**Friday, August 21st, 11:00 - 12:00**

**Algebraic analogues of coset posets and the low dimensional homology of Iwahori-Hecke algebras (Robin Sroka)**

Iwahori-Hecke algebras are a family of algebras associated to the choice of a Coxeter group and a commutative ring, an example being the group algebra of the Coxeter group. Every Coxeter group acts on a contractible space called the Davis complex, which is the geometric realization of a coset poset. In this talk, we introduce an algebraic analogue of the Davis complex for Iwahori-Hecke algebras and explain how it can be used to compute the low dimensional homology of Iwahori-Hecke algebras, interpreted as appropriate Tor-groups. The group algebra case recovers known computations of the low dimensional homology of Coxeter groups. This is a work in progress inspired by conversations with Hepworth, a paper of Boyd and their recent joint work.