The University of Sydney
School of Mathematics and Statistics  
Algebra Seminar  
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University of Sydney Algebra Seminar

Murray Elder

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.