The Tenth Barcelona Weekend in Group Theory took place on May 27th, 2015.

- 10:00
**Montse Casals**(Euskal Herriko Unibertsitatea, Universidad del País Vasco, Bilbao, Spain)Embeddability between right-angled Artin groups and its relation to model theory and geometry.

Abstract: In this talk we will discuss when a right-angled Artin group is a subgroup of another one, and explain how this basic algebraic problem may provide answers to questions in geometric group theory and model theory such as classification of right-angled Artin groups up to quasi-isometries and universal equivalence.

- 11:15
**Jim Howie**(Heriot-Watt University, Edinburgh, Scotland)Weight of groups and surgery on knots.

Abstract: The weight of a group G is the smallest integer n such that some subset of size n generates G as a normal subgroup of itself. In particular, any n-knot group has weight 1 for any n, as it is the normal closure of a meridian element. Hence, many problems arising from knot surgery reduce to questions about whether certain groups can have weight 1. I will talk about some interesting examples of such questions.

- 12:30
**Yash Lodha**(EPFL, Lausanne, Switzerland)A nonamenable finitely presented group of piecewise projective homeomorphisms.

Abstract: I will describe a finitely presented subgroup of Monod’s group of piecewise projective homeomorphisms of the real line. This provides a new example of a finitely presented group which is nonamenable and yet does not contain a nonabelian free subgroup. The example is torsion free and of type F∞. A portion of this is joint work with Justin Moore.

- 16:00
**Jianchun Wu**(Soochow University, Suzhou, China) - 17:15
**Lawrence Reeves**(Melbourne University, Melbourne, Australia)Coxeter groups and limit roots.

Abstract: Limit roots were recently introduced by Hohlweg, Labbe and Ripoll to study the asymptotic distribution of roots in a based root system. I will describe some recent work with Xiang Fu on the set of limit roots associated to an infinite Coxeter group.

Fixed subgroups are compressed in surface groups.

Abstract: For a compact surface $\Sigma$ (orientable or not, and with boundary or not) we show that the fixed subgroup, $\fix\B$, of any family $\B$ of endomorphisms of $\pi_1(\Sigma )$ is compressed in $\pi_1(\Sigma )$ i.e., $\rk(\fix\B)\leqslant \rk(H)$ for any subgroup $\fix\B\leqslant H\leqslant \pi_1(\Sigma )$. On the way, we give a partial positive solution to the inertia conjecture, both for free and for surface groups. We also investigate direct products, $G$, of finitely many free and surface groups, and give a characterization of when $G$ satisfies that $\rk(\fix \phi)\leqslant \rk(G)$ for every $\phi\in \aut(G)$.