Geometry and Topology of Artin Groups
Final Report Abstract
The project studied abstract group homomorphisms f : K → G from a topological group K to a discrete group G. The aim was to show that under favorable assumptions such a homomorphism either has an open kernel, or that the image is small in some group-theoretic sense. Here, K is a Cech complete topological group and G belongs to an interesting class of discrete groups, such as Artin groups, Coxeter groups, Gromov hyperbolic groups, or automorphism groups of these. We were able to settle this question in many cases. The main results are as follows. Theorem 1 Let K be a locally compact group and let f : K → G be an abstract homomorphism onto a discrete group G. If G does not contain the rationals (Q, +), the p-adic integers (Zp, +), or the Prüfer group Z(p∞), for any prime p, and if all torsion subgroups of G are artinian, then either f is continuous, or there is an open normal subgroup N ¢ K whose image f (N ) is torsion. Theorem 1 relies heavily on the structure theory of locally compact groups. The following result holds in the much more general context of Cech complete groups. Theorem 2 Let GΓ be a graph product and assume that there is a uniform upper bound n on the orders of the torsion elements in the vertex groups Gv. If f : K → GΓ is an abstract homomorphism from a Cech complete group K to GΓ, then f is continuous, or there is an open normal subgroup N ¢ K such that f (N ) is conjugate to a subgroup of G∆ ⊆ GΓ , for some clique ∆. Theorem 3 Let AΓ be a right-angled Artin group. Then both Aut(AΓ ) and Out(AΓ ) have finite index subgroups G such that every abstract homomorphism K → G is continuous, for every Cech complete group K. Keppeler’s 2024 PhD thesis contains many more results in this direction.
Publications
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Automatic continuity for groups whose torsion subgroups are small. Journal of Group Theory, 0(0).
Keppeler, Daniel; Möller, Philip & Varghese, Olga
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Automatic continuity in the Cech complete setting, PhD thesis, Münster 2024
D. Keppeler
