The Twelfth Barcelona Weekend in Group Theory took place on April 29th, 2017 at the Facultat de Matemàtiques i Estadística of the Universitat Politècnica de Catalunya.

- 10:00-11:00
**Colva Roney-Dougal**(University of St Andrews, Scotland).*Generating sets of finite groups* - 11:15-12:15
**Juan Souto**(CNRS and Université de Rennes 1, France).*Counting curves and the stable length of currents* - 12:30-13:30
**Peter Wong**(Bates College, Lewiston, Maine, USA).*Property $R_{\infty}$ and the Bieri-Neumann-Strebel invariant for finitely generated groups*

It is well known that generating sets for groups are far more complicated than generating sets for, say, vector spaces. The latter satisfy the exchange axiom, and hence any two irredundant sets have the same cardinality. According to the Burnside Basis Theorem, a similar property holds for groups of prime power order.

We define a new sequence of relations on the elements of a finite group, one for each positive integer r, where two elements of a finite group are equivalent if each can be substituted for the other in any r-element generating set. These relations become finer as r increases: we define a new numerical group invariant to be the value of r at which they stabilise. We are able to characterise this value of r for all soluble groups, and give upper and lower bounds for all finite groups.

The generating graph of a 2-generated finite group is a graph whose vertices are the group elements, and whose edges are the 2-element generating sets. Our new relations yield a precise description of the automorphism group of the generating graph of any finite soluble group with nonzero spread.

Fix a discrete and cocompact action of a surface group of genus $g\ge 2$ on a metric space. We study the asymptotic behavior of the number of curves on the surface of given type and with translation length at most L. For example we get that for any finite generating set of the surface group the limit $$\lim_{L\to\infty}\frac 1{L^{6g-6}}\vert\{\gamma\text{ simple with word length }\le L\}$$ exists and is positive. This result builds on previous work of Mirzakani and Erlandsson-Souto, together with the fact that the function which associates to each curve its stable length with respect to the action extends to a continuous and homogenous function on the space of currents. This last claim holds whenever we replace the surface group by an arbitrary hyperbolic group. This is joint work with Erlandsson and Parlier.

A group $G$ is said to have property $R_{\infty}$ if the number of $\varphi$-twisted conjugacy classes is infinite for every automorphism $\varphi$ of $G$. The study of twisted conjugacy classes originated from the classical theory of fixed point classes and thus the property $R_{\infty}$ has applications in topological fixed point theory. In this talk, I will describe how the BNS-invariant (and its related invariants) can be used to determine whether a finitely generated group has the property $R_{\infty}$.

This meeting is open to everybody, regardless of race, sex, religion, national origin, sexual orientation, gender identity, disability, age, pregnancy, immigration status, or any other aspect of identity.