Let $S_{g,p}^n$ be a compact oriented surface of genus $g$ with $p$ punctures and $n$ boundary components, and $\operatorname{Mod}(S)$ and $\operatorname{PMod}(S)$ be the mapping class group and the pure mapping class group of $S$, respectively. Suppose $n >0$ and fix a base point on a boundary. I know that there is a natural monomorphism $$\operatorname{Mod}(S_{g,p}^1)\to \operatorname{Aut}(\pi_1(S_{g,p}^1)),$$ and the image of $\operatorname{Mod}(S_{0,p}^1)$, which is isomorphic to the braid group, $\operatorname{PMod}(S_{0,p}^1)$, which is isomorphic to the pure braid group, and $\operatorname{Mod}(S_{g,0}^1)=\operatorname{PMod}(S_{g,0}^1)$ are known: respectively $$\{\phi \mid \phi(x_i)\textrm{ is a conjugate of }x_{\sigma(i)},\;\sigma\in \mathfrak{S}_p, \; \phi \textrm{ fixes the boundary}\},$$ $$\{\phi \mid \phi(x_i)\textrm{ is a conjugate of }x_i,\; \phi \textrm{ fixes the boundary}\},$$ $$\{\phi \mid \phi \textrm{ fixes the boundary}\},$$ where $x_i$ correspond the punctures, $\mathfrak{S}_p$ is the symmetric group.

How about the image of $\operatorname{PMod}(S_{g,p}^1)$, which is isomorphic to $\operatorname{Mod}(S_{g,0}^{p+1})/\langle\textrm{boundary Dehn twists}\rangle$), and $\operatorname{Mod}(S_{g,p}^1)$?

(Is the former $$\{\phi \mid \phi(x_i)\textrm{ is a conjugate of }x_i,\; \phi \textrm{ fixes the boundary}\}?)$$

At least, I want to know whether they are known or open.

  • $\begingroup$ I asked a similar question here and Allen Hatcher gave an enlightening answer: mathoverflow.net/a/308391/8103 $\endgroup$ – Mark Grant May 21 at 12:33
  • $\begingroup$ @MarkGrant I could not find any answer to my question. (Your question on the comment of Hatcher's reply is almost similar to my question, but there's no answer. Have you found the answer?) $\endgroup$ – qkqh May 22 at 15:36
  • $\begingroup$ We never found a reference, and so ended up changing the parameters of our problem. My question led to some email correspondence with another MO user, I'll email him to see if he made any progress. $\endgroup$ – Mark Grant May 22 at 19:57
  • $\begingroup$ @MarkGrant You mean that the problem is(was) unsolved, right? $\endgroup$ – qkqh May 22 at 21:04
  • $\begingroup$ I believe that there is no statement of Dehn-Nielsen-Baer for suraces with boundary and punctures in the literature. I also feel like someone who works a lot with mapping class groups should be able to figure out the answer, if they were sufficiently motivated. As I said, we shifted to looking at a slightly different problem, and the motiviation went away. (BTW, I'm talking about a project which led to this paper: arxiv.org/abs/1903.01916) $\endgroup$ – Mark Grant May 24 at 13:03

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