Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Asymptotic homology of the quotient of $PSL_2(\BR)$ by a modular group   Consider $ G:= PSL_2(\R)\equiv T^1\H^2$, a modular group $ \Gamma$, and the
homogeneous space $ \Gamma\sm G \equiv T^1(\Gamma\sm\H^2)$. Endow $ G $, and
then $ \Gamma\sm G $, with a canonical left-invariant metric, thereby equipping
it with a quasi hyperbolic geometry. Windings around handles and cusps of $
\Gamma\sm G $ are calculated by integrals of closed 1-forms of $ \Gamma\sm G $.
The main results express, in both Brownian and geodesic cases, the joint
convergence of the law of these integrals, with a stress on the asymptotic
independence between slow and fast windings. The non-hyperbolicity of $
\Gamma\sm G $ is responsible for a difference between the Brownian and geodesic
asymptotic behaviours, difference which does not exist at the level of the
Riemann surface $\Gamma\sm\H^2$ (and generally in hyperbolic cases).
Identification of the cohomology classes of closed 1-forms and with harmonic
1-forms, and equidistribution of large geodesic spheres, are also addressed.

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The categories of this paper are: cs
11
The categories of this paper are: hep-lat
8
The categories of this paper are: hep-ex
34
The categories of this paper are: chem-ph
15
The categories of this paper are: nucl-ex
0
The categories of this paper are: hep-ph
22
The categories of this paper are: chao-dyn
10
The categories of this paper are: q-bio
6
The categories of this paper are: astro-ph
29
The categories of this paper are: funct-an
19
The categories of this paper are: eess
24
The categories of this paper are: dg-ga
4
The categories of this paper are: cond-mat
27
The categories of this paper are: comp-gas
16
The categories of this paper are: stat
9
The categories of this paper are: nlin