RESEARCH GROUP

GEOMETRY

The preprint "On normal subgroups in automorphism groups" by Philip Möller and Olga Varghese has been recently uploaded to the arXiv!

Link to the paper: arXiv:2208.05677

Abstract:
We describe the structure of virtually solvable normal subgroups in the automorphism group of a right-angled Artin group Aut(AΓ). In particular, we prove that a finite normal subgroup in Aut(AΓ) has at most order two and if Γ is not a clique, then any finite normal subgroup in Aut(AΓ) is trivial. This property has implications to automatic continuity and to C∗-algebras: every algebraic epimorphism φ:L↠Aut(AΓ) from a locally compact Hausdorff group L is continuous if and only if AΓ is not isomorphic to Zn for any n≥1. Further, if Γ is not a join and contains at least two vertices, then the set of invertible elements is dense in the reduced group C∗-algebra of Aut(AΓ). We obtain similar results for Aut(GΓ) where GΓ is a graph product of cyclic groups. Moreover, we give a description of the center of Aut(GΓ) in terms of the defining graph Γ.

This summer, Marco Lotz will spend a couple of months in Australia visiting Anne Thomas at the University of Sydney.
Hopefully this will be a very productive research visit!

The following is new on the ArXiv:

Chimney retractions in affine buildings encode orbits in affine flag varieties

Elizabeth Milićević, Petra Schwer, Anne Thomas

This paper determines the relationship between the geometry of retractions and the combinatorics of folded galleries for arbitrary affine buildings, and so provides a unified framework to study orbits in affine flag varieties. We introduce the notion of labeled folded galleries for any affine building X and use these to describe the preimages of chimney retractions. When X is the building for a group with an affine Tits system, such as the Bruhat-Tits building for a group over a local field, we can then relate labeled folded galleries and shadows to double coset intersections in affine flag varieties. This result generalizes the authors' previous joint work with Naqvi on groups over function fields.