====== David Biddle (Binghamton) ======

====== On the Size and Connectivity of Graphs of Generating Sets of Finitely Generated Groups ======

===== Abstract for the Combinatorics and Algebra Seminars 2011 October 25 =====

Let G be a finitely generated group with minimal generating set of size d. For each t ≥ d let Γ<sub>t</sub> = Γ<sub>t</sub>(G) be the graph with vertex set V consisting of all generating t-tuples of elements of G and with edges ((g<sub>1</sub>, ..., g<sub>t</sub>), (g'<sub>1</sub>, ..., g'<sub>t</sub>)) if for some distinct i and j, g'<sub>i</sub> is g<sub>i</sub> multiplied on left or right by g<sub>j</sub><sup>±1</sup>, and all other g'<sub>k</sub> are the same as the corresponding g<sub>k</sub>.

Following work by Graham and Diaconis I examine connectivity properties of these graphs when G is abelian and when G is a small symmetric group. (For instance, |V (Γ<sub>3</sub>(Σ<sub>4</sub>))| = 10,080!!). Pictures will be provided free of charge.

I will relate the size and connectivity properties of these graphs to classic counting problems of Phillip Hall.


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