## Olivetti Club

A self-homeomorphism of a surface has a real number associated to it, called the 'dilatation', or the 'stretch factor'. These dilatations are algebraic units whose Galois conjugates lie in a specific annulus in the plane. Algebraic units that satisfy this property are called 'biPerron', but it is not known whether all biPerron numbers are dilatations of homeomorphisms. For a subset of the set of dilatations, we give a construction of a surface, and a homeomorphism of it, with the intended dilatation. To possibly shed further light on the relation between dilatations and biPerron units, we will give an asymptotic relation between the two, fixing the genus of the surface. And provide a few results in lower genera.

The picture above depicts the Galois conjugates of Lehmer's number, $\lambda$, a biPerron algebraic unit with minimal polynomial $x^{10}+x^9-x^7-x^6-x^5-x^4-x^3+x+1$.