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Hi, everyone.

I am not sure it is proper to ask the following question on here.

Let $S$ be a genus $g\geq 1$ surface with 2-puncture, i.e. genus $g$ closed surface with 2 points removed. And there is a compact surface $S_{1}\subset S$ such that $S-S_{1}$ consists of 2 once-punctured disk.

Now in Farb and Margalit' book "A primer on mapping class groups ", they defined the $Mod(S_{1})$ and $PMod(S)$.

My first question is

Is it true that we can consider treat $PMod(S)$ as a subgroup of $Mod(S_{1})$?

2 added 200 characters in body; deleted 184 characters in body

Hi, everyone.

I am not sure it is proper to ask the following question on here.

Let $S$ be a genus $g\geq 1$ surface with 2-puncture, i.e. genus $g$ closed surface with 2 points removed. And there is a compact surface $S_{1}\subset S$ such that $S-S_{1}$ consists of 2 once-punctured disk.

Now in Farb and Margalit' book "A primer on mapping class groups ", they defined the $Mod(S_{1})$ and $PMod(S)$.

My first question is

Is it true that we can consider $PMod(S)$ as a subgroup of $Mod(S_{1})$?

1

Pure Mapping class group and mapping class group

Hi, everyone.

I am not sure it is proper to ask the following question on here.

Let $S$ be a genus $g\geq 1$ surface with 2-puncture, i.e. genus $g$ closed surface with 2 points removed. And there is a compact surface $S_{1}\subset S$ such that $S-S_{1}$ consists of 2 once-punctured disk.

Now in Farb and Margalit' book "A primer on mapping class groups ", they defined the $Mod(S_{1})$ and $PMod(S)$.

My question is

Is it true that we can consider $PMod(S)$ as a subgroup of $Mod(S_{1})$?