## Topology & Geometric Group Theory Seminar

## Spring 2008

### 1:30 - 2:30, Malott 207

Tuesday, February 5

**Danny
Calegari**, Caltech

*Stable commutator length is rational in free groups*

For any group, there is a natural (pseudo)-norm on the vector space
B_{1} of real (group) 1-boundaries, called the stable
commutator length norm, which generalizes stable commutator length on
elements of the commutator subgroup. We show that in a free group,
the unit ball of this pseudo-norm, restricted to any finite
dimensional rational subspace of B_{1}, is a finite sided
rational polyhedron. As a corollary, we show that every element of a
free group has rational stable commutator length, and moreover every
element of the commutator subgroup rationally bounds an injective map
of a surface group into the free group.

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