Friday 2nd September, 12.05-12.55pm, Carslaw 275
Looking for short words in the first Grigorchuk group
The "first" Grigorchuk group as it is known is a finitely
generated group that is not finitely presented, although it does have
a nice recursive definition. Every element has finite order but it is
infinite, and its growth function f(n), which counts the number of
elements represented by a word of length n with respect to some
generating set, is between polynomial and exponential. Grigorchuk came
up with this group in the 80s, answering a question of Milnor as to
whether any group could have "intermediate" growth.
In my talk I will explain this group and discuss our current work to
describe the set of all "geodesic" or shortest length words with
respect to a four generator generating set. This set has some
intriguing structure, for instance, it contains an indexed language
such that no infinite subset is context-free.
The talk will require no background knowledge much, except for knowing
what a group presentation is. This is joint work with Mauricio
Gutierrez and Zoran Sunik.