韩国裸舞

Title: Effective Generalization of Hall's Theorem for Limit Groups and Cube Complexes

Abstract: A finitely generated group G is called subgroup separable if every finitely generated subgroup H of G is closed in the profinite topology on G (equivalently, there is a family of finite index subgroups of G intersecting in H). One of the initial motivations for studying residually finite groups and subgroup separable groups was McKinsey-Malcev algorithm solving the word problem in finitely presented residually finite groups. Recently, separability has played a crucial role in low-dimensional topology, namely in the resolutions of the Virtually Haken and Virtually Fibered conjectures.

A celebrated theorem of Marshall Hall implies that finitely generated free groups are subgroup separable and that each their finitely generated subgroup H is a retract of a finite-index subgroup K. It also states that K can be obtained from H by a series of free products with infinite cyclic groups. The first statement of the theorem was generalized by Wilton for limit groups. Haglund-Wise proved it for right-angled Artin groups when H is word quasiconvex. We generalize the second statement of Hall's theorem and prove that such K can be obtained from H by a series of certain HNN-extentions. This implies that if L is a right-angled Artin group, H a word quasiconvex 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 limit groups this implies similar polynomial bounds and the resolution of the Hanna Neumann conjecture.

These are joint results with K. Brown and A. Vdovina.

We will gather for teatime in the lounge after the talk and then we will go for dinner with Olga. Please let me know if you would be interested in joining for dinner.

christopher.karpinski [at] mail.mcgill.ca