We study fixed subgroups of automorphisms of right-angled Artin and Coxeter groups. If \phi is an untwisted automorphism of a RAAG, or an arbitrary automorphism of a RACG, we prove that Fix(\phi) is finitely generated and undistorted. Up to replacing \phi with a power, we show that Fix(\phi) is even quasi-convex with respect to the standard word metric. This implies that Fix(\phi) is separable and a special group in the Haglund-Wise sense. Some of our techniques are applicable in the more general context of Bowditch's coarse median groups. Based on arXiv:2101.04415.
Tim Clausen: Mock hyperbolic reflection spaces and Frobenius groups of finite Morley rank
Thursday, 22.04.2021 10:30 im Raum via Zoom
Joint work with Katrin Tent.
A Frobenius group is a group G together with a proper nontrivial malnormal
subgroup H. A classical result due to Frobenius states that finite Frobenius
groups split, i.e. they can be written as a semidirect product of a normal
subgroup and the subgroup H. It is an open question if this holds true for
groups of finite Morley rank, and the existence of a non-split Frobenius group
of finite Morley rank would contradict the Algebraicity Conjecture. We use
mock hyperbolic reflection spaces, a generalization of real hyperbolic spaces,
to study Frobenius groups of finite Morley rank.
We show that the involutions in a connected Frobenius group of finite Morley
rank and odd type form a mock hyperbolic reflection space. These spaces
satisfy certain rank inequalities and we conclude that connected Frobenius
groups of odd type and Morley rank at most 6 split.
Moreover, by using a construction from the theory of K-loops we show that if G
is a connected Frobenius group of degenerate type with abelian complement,
then G can be expanded to a group whose involutiuons almost form a mock
hyperbolic reflection space. The rank inequalities allow us to show structural
results for such groups. As a special case we get Frecon's theorem: There is
no bad group of Morley rank 3.
Goodwillie and Weiss developed a powerful homotopy theoretic technique for studying spaces of embeddings. For properly embedded arcs in a manifold of any dimension we give this theory a geometric flavour inspired by Vassiliev theory for classical knots. On one hand, in that classical case, this viewpoint can be used to prove some missing cases of Goodwillie-Klein connectivity estimates, and also partially confirm the conjecture that these invariants are universal Vassiliev. On the other hand, in higher dimensions we produce knotted families of arcs, which in joint work with Peter Teichner we apply to solve an open problem in 4-dimensional topology. In this talk I will outline these various cases, and try to point out a very exciting common thread.
Abstract: In recent years, Gromov proposed studying the geometry of positive scalar curvature ("psc'') via various metric inequalities vaguely reminiscent of classical comparison geometry. For instance, let $M$ be a closed manifold of dimension $n-1$ which does not admit a metric of psc. Then with respect to any Riemannian metric of scalar curvature $\geq n(n-1)$ on the cylinder $V = M \times [-1,1]$, the distance between the two boundary components of $V$ is conjectured to be at most $2\pi/n$. In this talk, we will discuss how to approach this and other related conjectures on spin manifolds via index-theoretic techniques. We will use variations of the spinor Dirac operator augmented by a Lipschitz potential and subject to suitable local boundary conditions. In the cases we consider, this leads to refined estimates involving the mean curvature of the boundary and to rigidity results for the extremal situation. Joint work with S. Cecchini.