## Topology & Geometric Group Theory Seminar

## Fall 2011

### 1:30 – 2:30, Malott 203

### Tuesday, October 25

**Peter
Kropholler**, University of Glasgow (visiting Cornell)

*Wilson's short proof of the Romanovskii-Wilson theorem*

The theorem says this: let m and n be natural numbers with m < n.
Suppose you have a group G which admits a presentation with n
generators and m relators. Then for any set Y of generators of G,
there is a subset of n-m elements of Y that freely generate a free
group of rank n-m. It is proved by using ordered groups and
embeddings in division rings to reduce it to the following statement
about finite dimensional vector spaces: if V is an n-dimensional
vector space and U is an m-dimensional subspace then any subset Y of
of V which spans V modulo U contains a subset of n-m vectors which
span a complement to U in V.

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