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A Primer on Mapping Class Groups (PMS-49) by Dan Margalit, Benson Farb

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Chapter Eight

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The Dehn–Nielsen–Baer Theorem

The Dehn–Nielsen–Baer theorem states that Mod(Sg) is isomorphic to an index 2 subgroup of the group Out(π1(Sg)) of outer automorphisms of π1(Sg). This is a beautiful example of the interplay between topology and algebra in the mapping class group. It relates a purely topological object, Mod(Sg), to a purely algebraic one, Out(π1(Sg)). Further, these are related via hyperbolic geometry!

8.1 STATEMENT OF THE THEOREM

We begin by defining the objects in the statement of the theorem.

Extended mapping class group. Let S be a surface without boundary. The extended mapping class group, denoted Mod±(S), is the ...

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