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Let $T_{g,n}$ be the Torelli group of a $n$-punctured surface $S=\overline{S}\setminus\{x_1,\ldots,x_n\}$, with $\overline S$ orientable, closed and of genus $g$. By definition, $T_{g,n}$ is the kernel of the following map $$ MCG_{g,n}\hookrightarrow MCG_g\rightarrow {\rm Aut}\big(H_1(\overline S,\mathbb Z), \wedge\big)\simeq Sp_g(\mathbb Z) $$ where $MCG_g$ (reps. $MCG_{g,n}$) stands for the mapping class group of $\overline{S}$ (reps. of $S$) and $\wedge$ denotes the symplectic intersection pairing on $H_1(\overline S,\mathbb Z)$.

Question: What is known about $T_{g,n}$ in the particular case when $g=1$ and $n>0$? Is this group finitely generated? Do we know an explicit system of generators? Etc.

A reference would be welcome.

Thanks for any help.

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  • $\begingroup$ The mapping class group of a torus is $SL(2,\Bbb{Z})$, so $T_1$ and $T_{1,n}$ are trivial. $\endgroup$
    – abx
    Feb 6, 2014 at 16:29
  • $\begingroup$ @abx: How are they trivial? As far as I understand, they must contain at least the subgroup of pure braids in $B_n$. $\endgroup$ Feb 6, 2014 at 18:19
  • $\begingroup$ Well, $T_1$ is trivial, and the OP claims that $T_{1,n}$ is contained in $T_1$. Maybe that part is wrong? $\endgroup$
    – abx
    Feb 6, 2014 at 19:15
  • $\begingroup$ You are right: the natural morphism $MCG_{g,n}\rightarrow MCG_g$ is surjective, not injective... $\endgroup$
    – Lucien
    Feb 6, 2014 at 22:08

1 Answer 1

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This group is finitely generated. This is classical, but in any case it follows easily from the much more general results in my paper

Putman, Andrew Cutting and pasting in the Torelli group. Geom. Topol. 11 (2007), 829–865.

which is devoted to understanding the relationships between various notions of the Torelli group on surfaces with multiple boundary components. The Torelli group you are interested in corresponds (in the notation of my paper) to the partition $\{\{b_1\},\ldots,\{b_n\}\}$ of the boundary components $b_1,\ldots,b_n$ of the surface.

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  • $\begingroup$ Thanks for the reference. It contains nice descriptions of set of generators of several Torelli groups $\mathcal I(S,P)$ for a partitioned surface $(S,P)$. But I can't find anything in it about finite generation/presentation of these groups. I certainly missed something... $\endgroup$
    – Lucien
    Feb 12, 2014 at 12:48
  • $\begingroup$ @LucienfromIHP : Theorem 1.2 of that paper shows that the Torelli groups you are looking at can be formed starting from the case $n=1$ (where the group in question is just $\mathbb{Z}$) as an iterated sequence of extensions by fundamental groups of unit tangent bundles of surfaces (which are obviously finitely presentable groups). They are thus finitely presentable. I guess that you want punctures instead of boundary components, so one would then have to quotient out by the copy of $\mathbb{Z}^n$ generated by Dehn twists about the boundary components. $\endgroup$ Feb 12, 2014 at 18:12
  • $\begingroup$ @ Andy Putman: thanks a lot! What about finite representation of these groups (for surfaces with punctures)? Are there some results when $n\geq 2$? $\endgroup$
    – Lucien
    Feb 13, 2014 at 13:33
  • $\begingroup$ @LucienfromIHP : I'm not sure exactly what you want here. They're residually finite (because they are subgroups of the mapping class group, which is residually finite). Completely classifying their finite quotients is probably hopeless. $\endgroup$ Feb 13, 2014 at 17:53

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