Marked mapping class group

Description

The marked mapping class group is a symbol used in lit_koberdaAsymptoticLinearityMapping2010 to denote a Mapping class group of a surface which fixes a point of the surface. Computationally it is equivalent to adding a puncture to the surface but conceptually it is used to give a mapping class group which acts on the non-free Fundamental group.

Definition

Let $S$ be a surface and $\ast$ a marked point. The marked mapping class group consists of the connected compontent of orientation preserving homeomorphisms fixing $\ast$: $MC{G}^1(S)={\pi }_0(Home{o}^{+}(S),\ast )$ The point $\ast$ is fixed by the isotopies as well.

Characerisation by Automorphisms

The marked mapping class group acts naturally and faithfully on the Fundamental group with base point $\ast$. From this we deduce that the marked mapping class group can be identified with a subgroup of $Aut({\pi }_1(S,\ast ))$

Properties

Forgetting the base point

Let $S$ be a surface with marked point $\ast \in S$. By forgetting the base point we get a surjective map to the Mapping class group $MC{G}^1(S)\to MCG(S)$

Examples

Example