Let $T$ be the 2-dimensional torus and let $S$ be $T$ minus one point. Then Birman exact sequence of mapping class groups becomes an isomorphism $$ \beta: Map(S)\to Map(T)=GL(2, {\ …

Is the following true? "The conjugacy classes of two homeomorphisms of the n-times punctured disk have isotopic representatives iff the associated mapping tori are homeomorphic." …

Hi everyone. Let $S$ be a closed surface with genus at least 3, $\alpha, \beta$ be the two vertices of curve complex of $S$ such that $d_{\mathcal {C}(S)}(\alpha, \beta)\geq 3$. …

Let $B_{2g+1}$ be the Artin braid group on $2g+1$ strands. There is a symplectic representation $\rho: B_{2g+1} \rightarrow Sp_{2g}(\mathbf{Z})$ called the "hyperelliptic repr …

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 poin …

Hi, everyone. Assume $S$ is a genus at least 2 orientable closed surface. And there is a simplical complex defined on $S$ called Curve complex. It is well known that any autom …

(The question has been edited. It was pointed out in the comments that $\Gamma_G$ could be a surface group, thought of as a finite extension of another surface group, in which cas …

If $S$ is the surface illustrated below, do the Dehn twists about the red curves generate the mapping class group $\operatorname{MCG}(S,\partial S)$?

I am curious to see if anyone knows a proof of the Nielsen-Thurston classification of mapping classes that does not depend on results in Teichmuller theory. From a naive point o …

Assume $V$ be a genus larger than 1 handlebody, $S=\partial_{+} V$. Denote $N$ be the normal closure of $MCG(V)$ in $MCG(S)$. Is there any material related to the quotient group …

Let $N_{k,r}$ be a non-orientable surface of genus $k$ (i.e the connected sum of $k$ projective planes) and with $p\geq 1$ punctures. I'm looking at ideal triangulations of the s …

I have a question about a proof that I am reading in "A primer on Mapping Class Groups" by Farb and Margalit. Let $a$ be a simple closed curve in a compact surface $S$ (possibly w …

Let $B_n$ be the braid group on $n$ strands. It's a theorem of Daan Krammer and Stephen Bigelow that there is a a faithful representation $$B_n \to GL_{n \choose 2} \mathbb Z[t^{ …

Let $V$ be a compressionbody. Annulus and disk complex $AD(V)$ is defined to be: Vertex: An istopy class of spanning annulus or an essential disk. Place an edge between two ve …

QUESTION: Let $g \geq 4$, $S(g)$ be the fundamental group of the genus $g$ surface, and $G$ be finitely generated (the number of generators $\leq 3$) group with abelianization of …