Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Universal commensurability augmented Teichm\"uller space and moduli
  space   It is known that every finitely unbranched covering
$\alpha:\widetilde{S}_{g(\alpha)}\rightarrow S$ of a compact Riemann surface
$S$ with genus $g\geq2$ induces an isometric embedding $\Gamma_{\alpha}$ from
the Teichm\"uller space $T(S)$ to the Teich\"uller space
$T(\widetilde{S}_{g(\alpha)})$. Actually, it has been showed that the isometric
embedding $\Gamma_{\alpha}$ can be extended isometrically to the augmented
Teichm\"{u}ller space $\widehat{T}(S)$ of $T(S)$. Using this result, we
construct a directed limit $\widehat{T}_{\infty}(S)$ of augmented Teichm\"uller
spaces, where the index runs over all finitely unbranched coverings of $S$.
Then, we show that the action of the universal commensurability modular group
$Mod_{\infty}(S)$ can extend isometrically on $\widehat{T}_{\infty}(S)$.
Furthermore, for any $X_{\infty}\in T_{\infty}(S)$, its orbit of the action of
the universal commensurability modular group $Mod_{\infty}(S)$ on the universal
commensurability augmented Teichm\"uller space $\widehat{T}_{\infty}(S)$ is
dense. Finally, we also construct a directed limit $\widehat{M}_{\infty}(S)$ of
augmented moduli spaces by characteristic towers and show that the subgroup
$Caut(\pi_{1}(S))$ of $Mod_{\infty}(S)$ acts on $\widehat{T}_{\infty}(S)$ to
produce $\widehat{M}_{\infty}(S)$ as the quotient.

19
The categories of this paper are: eess
5
The categories of this paper are: gr-qc
22
The categories of this paper are: chao-dyn
3
The categories of this paper are: physics
9
The categories of this paper are: nlin
10
The categories of this paper are: q-bio
16
The categories of this paper are: stat
1
The categories of this paper are: cs
13
The categories of this paper are: quant-ph
27
The categories of this paper are: comp-gas
20
The categories of this paper are: acc-phys
8
The categories of this paper are: hep-ex
14
The categories of this paper are: math-ph
23
The categories of this paper are: solv-int
37
The categories of this paper are: supr-con
18
The categories of this paper are: econ