We will have a double session this week.
Tuesday January 12th, 15:00, CRM Small Room
Speaker: Francesco Matucci (University of Virginia)
Title: Bounding the residual finiteness of free groups.
Abstract: We find a lower bound to the size of finite groups detecting a given word in the free group. More precisely, by using results on the length of shortest identities in finite simple groups we construct a word w of length n in non-abelian free groups with the property that w is the identity on all finite quotients of size ~ n^{2/3} or less. This improves on a previous result of Bou-Rabee and McReynolds quantifying the lower bound of the residual finiteness of free groups. This is joint work with Martin Kassabov.
Second talk, 16:00
Speaker: Sean Cleary (The City College of New York and the CUNY Graduate Center)
Title: Weak almost convexity and tame combing conditions
Abstract: Cannon introduced the notion of almost convexity for a Cayley graph of a group, developing effective algorithms for understanding the geometry of the Cayley graph. Though wide classes of groups are almost convex, there are a number of weaker notions of almost convexity satisfied by yet more groups. Mihalik and Tschantz introduced the notion of tame 1-combings for Cayley complexes, and there are connections between the very strongest notions of tame combings and the strongest notions of almost convexity. We answer questions about possible further connections between these two notions by showing that there are groups which satisfy some of the strongest tame combing conditions which do not satisfy even the weakest non-trivial almost convexity conditions. Examples include some Baumslag-Solitar groups and Thompson's group F. This is joint work with Susan Hermiller, Melanie Stein and Jennifer Taback.
