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In fact, much more is known: $Diff(\Sigma(p,q,r))\simeq Isom(\Sigma(p,q,r))$ when $\frac1p+\frac1q+\frac1r <1$ by a result of McCullough and Soma. The metric is a homogeneous metric on $\Sigma(p,q,r)$ …

No, this is false. The point is that if you take an irreducible periodic automorphism $f$, there is no disjointly embedded collection of curves of $\mathcal{C}(S)$ which is left invariant by $f$ up to …

As I said in the comment above, knots in $D^2\times S^1$ which have a Dehn filling giving $D^2\times S^1$ were classified by Gabai and Berge. Gabai proved that knots in $D^2\times S^1$ giving back $D^ …

Just an observation: consider the subgroup $\mathcal{G}_S\leq Mod(S)$ of the mapping class group of $S$ represented by a homeomorphisms $h:S\to S$ which extend to an orientation-preserving homeomorphi …

Yes, this is true - there are many ways to prove it, but I'll hit it with a hammer. Let $C(\Sigma)$ denote the curve complex of $\Sigma$. Suppose not, then $f$ would map every curve $[c]$ in $\mathcal …

Your question may be recast as: Is there an $N(n)$ such that for any integral matrix $A\in SL(n,\mathbb{Z})$, there is $k\leq N(n)$ with $tr(A^k)>2 $ (with a few small exceptional cases)? (this is n …

This works for $g=2$, but it’s a very special case. $Out(F_2)\cong GL_2( \mathbb{Z}) \cong Mod_1 \cong Mod_{1,1}$, the mapping class group of a pointed torus. This is realized by the linear action of …

A result of Behrstock and Minsky (cf. Hamenstadt too) implies that the rank of mapping class groups is the maximal rank of abelian subgroups, which is $3g+p-3$ for a connected hyperbolic surface of ge …

I'll address the question in your comments, can one determine a presentation for a subgroup of the mapping class group generated by Dehn twists? Given a bunch of simple closed curves on a surface, the …

As some further evidence for a positive answer to your question, there is a paper of Cantat and Loray that proves a relative version of this for mapping class group the 4-holed sphere (rel boundary). …

Edit: This answer was wrong - I misremembered the result. See Allen Hatcher's answer. Of course, the references in the second paragraph still stand. Those two operations give trivial outer automorphi …

It's finite index by Margulis' normal subgroup theorem. Since $H \lhd P_{2g+1}$, then $\rho(H)\lhd \rho(P_{2g+1})$. Since $\rho(P_{2g+1})$ is finite index in $\rho(B_{2g+1})=\Gamma(2)$ (I'm taking y …

For the first part of your question, there are faithful representations $\rho: \Gamma_g \to SO(3)$. One way to see this is by taking an arithmetic realization of the surface $\Sigma_g$, giving a faith …

Dave Gabai proved that the mapping class group of a closed hyperbolic 3-manifold is isomorphic to its isometry group. For a hyperbolic knot $K$ without any symmetries, for large enough $n$, $S^3_{1/n} …

(1) Bridson showed that if a mapping class group of a surface (of genus at least 3) acts on a CAT(0) space, then Dehn twists act as elliptic or parabolic elements. This implies that the mapping class …