On the first cohomology group of cofinite subgroups in surface mapping class groups

Following the well-known analogy between arithmetic groups and surface mapping class groups Ivanov asked whether the first cohomology group of any subgroup of finite index in a surface mapping class group must be trivial. In this note, we establish, as our first result, an affirmative answer to Ivanovʹs question, provided the surface in question has genus at aleast 3, and the subgroup of finite index contains the Torellli group. Secondly, we show that our first result does not hold for any surface of genus 2. This second result established, in particular, a negative answer to Ivanovʹs question for any surface of genus 2.