The Thirteenth Barcelona Weekend in Group Theory took place on April 27th and 28th, 2018 at the Centre de Recerca Matemàtica and the Facultat de Matemàtiques i Estadística of the Universitat Politècnica de Catalunya.
Friday April 27th, 2018, room A2, Centre de Recerca Matemàtica.
- 15:00-16:00 Jordi Delgado (Euskal Herriko Uniberstsitatea/Universidad del País Vasco, Bilbao, Spain). A geometric description for subgroups of abelian-by-free groups
- 16:15-17:15 Luis Mendoça (Universidad de Zaragoza, Spain). Σ-invariants of wreath products
- 17:30-18:30 Joan Porti (Universitat Autònoma de Barcelona, Spain). Anosov groups
We extend to abelian-by-free groups the classical Stallings theory describing subgroups of the free group as automata.
I will briefly review the basic techniques and implications for the free-case, and discuss the generalization to abelian-by-free groups by admitting abelian labels in the automaton.
This approach allows us to solve the membership problem and provides new insight on some other algorithmic problems, including the conjugacy problem (known to be unsolvable in this context), and the intersection problem (for which any Hanna Neumann-like bound is denied).
Joint work with Enric Ventura.
The Σ-invariants of groups, also called BNS or BNSR-invariants, are some mysterious geometric objects that contain a lot of information on the inheritance of finiteness properties by subgroups. It is usually a very hard task to describe the invariants for any specific class of groups. We will discuss the low dimensional invariants of (restricted permutational) wreath products of groups. By low dimensional we mean the invariants related to the inheritance of finite generation and finite presentability. The results may be applied to the study of the Reidemeister number of automorphisms of wreath products in some specific cases.
Anosov groups are the higher rank one analogue of convex cocompact subgroups of isometries of hyperbolic space. I will discuss several characterizations of those groups, both in hyperbolic space and in higher rank symmetric spaces. In particular I will focus in the point of view of coarse geometry. This is joint work with Misha Kapovich and Bernhard Leeb.
- 10:30-11:30 Andrei Jaikin-Zapirain (Universidad Autónoma de Madrid, Spain). The Strengthened Hanna Neumann conjecture for surface groups
- 11:45-12:45 Amnon Rosenmann (Graz University of Technology, Graz, Austria). Dependence over subgroups of free groups
The Hanna Neumann conjecture is a statement about the rank of the intersection of two finitely generated subgroups of a free group. The conjecture was posed by Hanna Neumann in 1957. In 2011, a strengthened version of the conjecture was proved independently by Joel Friedman and by Igor Mineyev.
In my talk I will introduce a variation of the Strengthened Hanna Neumann condition, which instead of ranks involves the Euler characteristic. Then I will prove the the surface groups satisfy this condition. As a consequence, we obtain that the original Strengthened Hanna Neumann conjecture holds not only in free groups but also in non-abelian surface groups. The talk is based on a joint work with Yago Antolin.
An element g of a free group F depends on a subgroup H < F if it satisfies a non-trivial univariate equation over H. Equivalently, when H is finitely-generated, the rank of
