# Dehn-Nielsen-Baer Theorem for surfaces with boundary and punctures

Let $S=S_{g,b}$ be a compact orientable surface with genus $g$ and $b$ boundary components, such that $\chi(S)=2-2g-b<0$. Let $Q=\{x_1,\ldots , x_n\}$ be a set of $n$ distinguished points in the interior of $S$.

Define ${\rm Mod}(S,\{Q\})$ to be the mapping class group of orientation-preserving self-homeomorphisms of $S$ which fix the boundary $\partial S$ pointwise and preserve $Q$ setwise, modulo isotopies of the same type. I believe this is the same as the mapping class group ${\rm Mod}(S-Q)$ of the punctured surface with boundary $S-Q$.

Now fixing a basepoint $d\in \partial S$, there are homomorphisms $${\rm Mod}(S,\{Q\})\to {\rm Aut}(\pi_1(S-Q,d))\to {\rm Out}(\pi_1(S-Q,d)).$$

Is there a version of the Dehn-Nielsen-Baer theorem which states that the above composition is isomorphic onto the subgroup ${\rm Out}^*(\pi_1(S-Q,d))$ of ${\rm Out}(\pi_1(S-Q,d))$ consisting of outer automorphisms which preserve the individual conjugacy classes represented by the components of $\partial S$, and permute the conjugacy classes represented by simple closed curves around the punctures?

We have looked in the following sources, which are all very helpful but seem to only treat the cases of boundary and punctures separately:

Farb, Benson; Margalit, Dan, A primer on mapping class groups, Princeton Mathematical Series. Princeton, NJ: Princeton University Press (ISBN 978-0-691-14794-9/hbk; 978-1-400-83904-9/ebook). xiv, 492 p. (2011). ZBL1245.57002.

Ivanov, Nikolai V., Mapping class groups, Daverman, R. J. (ed.) et al., Handbook of geometric topology. Amsterdam: Elsevier. 523-633 (2002). ZBL1002.57001.

Zieschang, Heiner; Vogt, Elmar; Coldewey, Hans-Dieter, Surfaces and planar discontinuous groups. (Poverkhnosti i razryvnye gruppy)., Moskva: Nauka. 688 p. (1988). ZBL0701.57001.

Boldsen, Soren, Different versions of mapping class groups of surfaces, https://arxiv.org/abs/0908.2221

The issue here is Dehn twists along curves parallel to the circles of $\partial S$. These usually generate infinite cyclic subgroups of ${\rm Mod}(S,Q)$, the only exceptions being when $S$ is a disk and $Q$ is empty or a single point. If one chooses a basepoint in $\partial S$ then these Dehn twists induce the identity on $\pi_1(S-Q)$ except for a twist along a curve parallel to the component of $\partial S$ containing the basepoint, which induces the inner automorphism which ccnjugates $\pi_1(S-Q)$ by the loop given by this component of $\partial S$. Thus the composition ${\rm Mod}\to {\rm Aut}\to {\rm Out}$ will usually have a nontrivial kernel.
• Hi Allen, thanks for your answer, it is very revealing. May I ask some follow up questions? (1) Is the kernel generated by these Dehn twists? and (2) do we have surjectivity onto ${\rm Aut}^*\pi_1(S-Q,d)$ (automorphisms fixing the individual conj. classes of boundary curves and permuting conj. classes of curves around the punctures)? – Mark Grant Aug 16 '18 at 9:34
• @MarkGrant For (2), the surjectivity, at least, we should add a condition that the automorphisms fix the component of $\partial S$ containing the basepoint. – qkqh May 21 '20 at 18:08
• @qkqh: I was assuming in the question that automorphisms fix the whole of $\partial S$ pointwise. – Mark Grant May 25 '20 at 9:27