WoG 2024 Talk 5.3: Penelope Azuelos - A geometric criterion for detecting virtual fiber subgroups

Speaker: Penelope Azuelos
Institution: University of Bristol
Website: https://penelopeazuelos.com/

Title: A geometric criterion for detecting virtual fiber subgroups
Abstract: A finitely generated subgroup H of a finitely generated group G is a virtual fiber subgroup if G admits a finite index subgroup which surjects onto the integers and the kernel has finite index in H. If the Schreier graph of G/H is in some way geometrically similar to the integers (e.g. it has two ends, it's a quasi-line...) then when is H a virtual fiber subgroup? Answers to this question naturally depend on the geometric property imposed on the Schreier graph and answers in the case where H has two relative ends and two filtered ends were provided by Houghton and Kropholler-Roller respectively. We will consider this question under a different condition, requiring instead that the Schreier graph is "narrow" (e.g. has linear growth, is a finitely ended quasi-tree) and has at least two ends and, under this hypothesis, give a characterisation of virtual fiber subgroups. Time permitting, we will also discuss examples of finitely generated subgroups whose Schreier graphs are quasi-lines but which fail to be virtual fibers.