Michal Kotrbčík: Genus of groups containing Z_3 factors

Pozor, ura je spremenjena. S seminarjem začnemo točno ob 11h (11.00)

Povzetek. The genus of a group Gamma is the minimum genus of a Caley graph of Gamma. For abelian groups not containing a Z_3 factor in their canonical form it is possible to exactly determine their minimum genus. We investigate the genus of G_n, the cartesian product of n triangles, which is a Cayley graph of (Z_3)^n. Using a lifting method we present a general construction of a low-genus embedding of G_n using a low-genus embedding of G_{n-1} satisfying some additional conditions. Our method provides currently the best upper bound on the genus of G_n for all n>=5. We show that the genus of G_4 is at least 30. Additionally, we discuss algorithmic aspects of the problems of determining the minimum genus and the complete embedding distribution of a graph. We report results obtained by a computer search which include improving the upper bound on the genus of G_4 to 39, complete genus distribution of G_2, and more than 200 nonisomorphic genus embeddings of G_3.

This is a joint work with T. Pisanski.