# Genus of groups containing Z 3 factors (Seminar DM)

Genus of groups containing Z_3 factors

Michal Kotrbčík

Torek, 8. november 2011, od 11.00-12.00, Plemljev seminar, Jadranska 19

Povzetek:

The genus of a group Γ is the minimum genus of a Caley graph of Γ. For abelian groups not containing a Z3 factor in their canonical form it is possible to exactly determine their minimum genus. We investigate the genus of Gn, the cartesian product of n triangles, which is a Cayley graph of (Z3)n. Using a lifting method we present a general construction of a low-genus embedding of Gn using a low-genus embedding of Gn − 1 satisfying some additional conditions. Our method provides currently the best upper bound on the genus of Gn for all $n\ge 5$. We show that the genus of G4 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 G4 to 39, complete genus distribution of G2, and more than 200 nonisomorphic genus embeddings of G3.

This is a joint work with T. Pisanski.