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Groups, Geometry and Topology

All lectures take place in V5-148 and U2-135 in Bielefeld.

Organizers: Barbara Baumeister, Kai-Uwe Bux, Linus Kramer

Talks:

10:15 - 11:15 Ilaria Castellano (Bielefeld) (in V5-148)Given a finitely generated group, thanks to the Stallings' decomposition theorem, it may be possible to split the group over a finite subgroup into pieces which in turn could split again. If this process terminates, i.e., it is not possible to continue splitting the group ad infinitum, the group is said to be accessible. In this talk I will first illustrate how the notion of accessibility carries over to the realm of totally disconnected locally compact (=t.d.l.c.) groups, and then move the attention to the t.d.l.c. groups of rational discrete cohomological dimension one. For these groups one can exploit the Euler-Poincaré characteristic to obtain an analogue of Linnell's theorem on the accessibility of finitely generated groups whose finite subgroups have bounded order.

11:15 - 11:45 Coffee break

11:45 - 12:45 Marco Amelio (Muenster) (in V5-148)

Until recently, the existence of non-split sharply 2-transitive groups (i.e., sharply 2-transitive groups without a non-trivial normal abelian subgroup) was an open problem. The first examples were produced by Rips, Segev and Tent in 2017 and by Rips and Tent in 2019. It is possible to associate to every sharply 2-transitive group a characteristic, that is either 0 or a positive prime number. The first of these examples have characteristic 2, while the other ones have characteristic 0, leaving the problem open for odd characteristics. In this talk, I will give some background on sharply 2-transitive groups and a rough overview on how we adapted the construction used for characteristic to 0 to odd characteristic, using methods of geometric small cancellation.

12:45 - 14:15 Lunch break

14:15 - 15:15 Tomas Reunbrouck (Bielefeld) (in U2-135)

In many interesting enumeration problems, generating functions can be used to leverage methods from complex and p-adic analysis. The classical example of this method is the use of the Poincaré series associated with an integer polynomial f. This is a (formal) power series, the i-th coefficient of which enumerates f's solutions modulo p^i for a fixed prime number p. This power series is known to be rational - that is, a quotient of two polynomials - which implies the existence of strong regularities in the sequence of coefficients. In particular, its poles describe the exponential growth rates present therein. In asymptotic group theory, we try to replicate this success when counting, for example, finite index subgroups of a given group. To do this, we use the local subgroup zeta function, which has properties similar to a Poincaré series, such as rationality. Moreover, its poles are important asymptotic invariants, which connect to the group's structure in quite subtle ways. Using p-adic analysis, we attempt to deepen our understanding of this relationship.

15:15 - 15:45 Coffee break

15:45 - 16:45 Raquel Murat García (Münster) (in U2-135)

For a compact group G with two closed subgroups P, Q such that P lies inside Q, it is known that the natural projection between the quotients G/P and G/Q is a Serre fibration, i.e., it has the homotopy lifting property with respect to every CW-complex. In this talk I will explain the steps taken to try to generalize this theorem to locally compact groups, although we will need to put some further conditions on its subgroups. I will prove that for a group G with a closed locally compact almost connected subgroup A, the projection of G onto G/A is a Serre fibration.