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# Tenth Barcelona Weekend in Group Theory (2015)

Speakers: Montse Casals (University of the Basque Country), Jim Howie (Heriot-Watt University), Yash Lodha (EPFL, Lausanne), Lawrence Reeves (Melbourne University, Melbourne, Australia), Jianchun Wu (Soochow University, Suzhou, China).

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

The Tenth Barcelona Weekend will coordinate its activities with the Joint Meeting of the Edinburgh Mathematical Society with the Catalan Mathematical Society, which takes place on May 28th-30th, 2015. The Joint Meeting will have a Special Session on Geometric Group Theory and a plenary talk by Enric Ventura. All together, it will be three days of group theory, with twelve talks on the subject (plus many talks on other subjects at the EMS-SCM meeting). Everyone is encouraged to attend both meetings.

Here is the program for the Tenth Barcelona Weekend. Talks will take place in FME-UPC, Pau Gargallo 5, Barcelona, Room 003.

• 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.

• Lunch
• 16:00 Jianchun Wu (Soochow University, Suzhou, China)
• 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 $\phi\in \aut(G)$.

• 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.

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