Arnold's seminar, Moscow University
February 28, 2006

Sergei Anisov (Utrecht, the Netherlands)
Convex hulls of $Z_p$-orbits in $R^4$

Given the p-element cyclic group Z_p acting on C^2=R^4
by g(z,w)=(\xi z, \xi^q w) (where g is a generator of Z_p,
numbers p and q are coprime, and \xi is a primitive p-th root
of unity), we describe the convex hull of a generic orbit.
"Genericity" means that z,w are nonzero and 1<q<p-1: otherwise
the orbits are flat regular p-gons. We show that this polytope
is simplicial (though it is not true that a hyperplane never
contains more than 4 of its vertices) and the number of its
facets equals p(E(p,q)-3), where
        E(p,q)=n_1+n_2+...+n_k,
where n_1+1/(n_2+1/(...+1/n_k)...) is the continued fraction
expansion of p/q. Moreover, the combinatorial type of this
polytope can be completely described in terms of the Euclid
algorithm and can be visualized in terms of the Farey tesselation
of hyperbolic plane.

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