# On the procongruence completion of the Teichm\"uller modular group

Research paper by **Marco Boggi**

Indexed on: **18 Jan '13**Published on: **18 Jan '13**Published in: **Mathematics - Algebraic Geometry**

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#### Abstract

For $2g-2+n>0$, the Teichm\"uller modular group $\Gamma_{g,n}$ of a compact
Riemann surface of genus $g$ with $n$ points removed $S_{g,n}$ is the group of
homotopy classes of diffeomorphisms of $S_{g,n}$ which preserve the orientation
of $S_{g,n}$ and a given order of its punctures. Let $\Pi_{g,n}$ be the
fundamental group of $S_{g,n}$, with a given base point, and $\hat{\Pi}_{g,n}$
its profinite completion. There is then a natural faithful representation
$\Gamma_{g,n}\hookrightarrow Out(\hat{\Pi}_{g,n})$. The procongruence
completion $\check{\Gamma}_{g,n}$ of the Teichm\"uller group is defined to be
the closure of the Teichm\"uller group $\Gamma_{g,n}$ inside the profinite
group $Out(\hat{\Pi}_{g,n})$.
In this paper, we begin a systematic study of the procongruence completion
$\check{\Gamma}_{g,n}$. The set of profinite Dehn twists of
$\check{\Gamma}_{g,n}$ is the closure, inside this group, of the set of Dehn
twists of $\GG_{g,n}$. The main technical result of the paper is a
parametrization of the set of profinite Dehn twists of $\check{\Gamma}_{g,n}$
and the subsequent description of their centralizers. This is the basis for the
Grothendieck-Teichm\"uller Lego with procongruence Teichm\"uller groups as
building blocks.
As an application, we prove that some Galois representations associated to
hyperbolic curves over number fields and their moduli spaces are faithful.