Pure Maths Colloquium: Stable commutator length in free groups
5 December 2011 16:00 in CM221
Stable commutator length (scl) answers the question: 'what is the simplest surface in a given space with prescribed boundary?' where 'simplest' is interpreted in topological terms. This topological deﬁnition is complemented by several equivalent deﬁnitions - in group theory, as a measure of non-commutativity of a group; and in linear programming, as the solution of a certain linear optimization problem. On the topological side, scl is concerned with questions such as computing the genus of a knot, or ﬁnding the simplest 4-manifold that bounds a given 3-manifold. In free groups, stable commutator length turns out to be a rich and mysterious invariant, with connections to dynamics, hyperbolic surfaces, and the geometry of integral polyhedra. We will discuss some of the phenomena that arise, and the connections to various areas of mathematics.
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