Geometric Group Theory (2014 REU)

A central idea of geometric group theory is to view groups as geometric objects with the hope of using geometric reasoning to discover algebraic facts. One way to view a finitely generated group \(G\) geometrically is to construct its Cayley graph with respect to a finite generating set \(S\). This graph has one vertex for every group element and an edge connecting elements \(g\) and \(h\) if \(g^{−1}h\) or \(h^{−1}g\) belongs to \(S\). The Cayley graph can be used to define a metric, \(d\), (the “word metric”) on a group by setting \(d(g,h)\) equal to the distance in the Cayley graph between the vertices corresponding to \(g\) and \(h\).

Familiar examples of Cayley graphs are the group of integers, \(\mathbb Z\), viewed as the integer points of the real line and the group \(\mathbb Z \times \mathbb Z \) viewed as the integer lattice points in the plane connected by horizontal and vertical lines. Cayley graphs can be beautiful but very complicated objects. Common groups whose Cayley graphs are still mysterious include mapping class groups, groups of automorphisms of free groups and Thompson’s groups \(F\), \(T\) and \(V\).

The geometric group theory group will study properties of the word metric on various classes of groups. Properties we may study include: existence of “dead ends”, almost convexity and uniform embedding into Hilbert space.

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