Group theory seminar Jaikin-Zapirain/Marquis/Kharlampovich

For this seminar we will have the pleasure of listening to three talks:

14.00 – 14.45 Andrei Jaikin-Zapirain (UAM), « Compressed subgroups in free groups are inert »

15.00 – 15.45 Timothée Marquis (UCL), « Amalgams of rational unipotent groups and residual nilpotence ».

16.15 – 17.00 Olga Kharlampovich (CUNY), « Quantification of separability of cubically convex-cocompact subgroups of RAAGs via representations ».

 Andrei Jaikin-Zapirain  « Compressed subgroups in free groups are inert ». 

 Let F be a free group. A finitely generated subgroup H is called compressed in F if it is not contained in a subgroup of F of smaller rank than H, and it is called inert in F if H ∩ U is compressed in U for any subgroup U of F. In my talk, I will show that compressed subgroups are also inert. The solves a conjecture of Dicks and Ventura from 1996.

Timothée Marquis , « Amalgams of rational unipotent groups and residual nilpotence ».

 Given a group property (P), a group G is called residually (P) if every nontrivial element of G has a nontrivial image in some quotient of G that satisfies (P). The study of residual properties of graphs of groups has a long and rich history, originating from Magnus’ theorem that free groups are residually torsionfree nilpotent. In this talk, I will start by reviewing a few key results of this history, before presenting an intriguing phenomenon concerning the residual nilpotence of certain amalgams of rational unipotent groups. Joint work with Pierre-Emmanuel Caprace. 

Olga Kharlampovich,   « Quantification of separability of cubically convex-cocompact subgroups of RAAGs via representations« .

 We answer the question asked by Louder, McReinolds and Patel and prove the following statement. Let L be a RAAG, H a cubically convex-cocompact subgroup of L, then there is a finite dimensional representation of L that separates the subgroup H in the induced Zariski topology. As a corollary, we establish a polynomial upper bound on the size of the quotients used to separate H in L. This implies the same statement for a virtually special group L and, in particular, a fundamental group of a hyperbolic 3-manifold. For any finitely generated subgroup H of a limit group L we prove the same results and, in addition, show that there exists a finite-index subgroup K containing H, such that K is a subgroup of a group obtained from H by a series of extensions of centralizers and free products with infinite cyclic group. If H is non-abelian, the K is fully residually H. A corollary is that a hyperbolic limit group satisfies the Geometric Hanna Neumann conjecture. These are joint results with K. Brown and A. Vdovina.

- Séminaire de théorie des groupes à l’ENS