THE KAZHDAN PROPERTY OF THE MAPPING CLASS GROUP OF CLOSED SURFACES AND THE FIRST COHOMOLOGY GROUP OF ITS COFINITE

SUBGROUPS

FERAYDOUN TAHERKHANI

Abstract. We show that the mapping class group of a closed surface of genus 2 does not satisfy the Kazhdan property by constructing subgroups of finite index having a nonvanishing first cohomology group. We also construct some subgroups of finite index in the mapping class group of a genus 3 surface and calculate their first cohomology groups, which all turn out to be trivial. Most of the calculations have been carried out by computer using GAP.