Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Multi-geodesic tessellations, fractional Dehn twists and uniformization
  of algebraic curves   Identifying parallel sides of a collection of Euclidean polygons yields a
flat surface with cone points of angles multiples of 2 pi, naturally a compact
Riemann surface but also an algebraic curve, and a hyperbolic surface. In
general two different metrics on a surface have no geodesic arcs in common, but
in special cases the surface is decomposed into polygons geodesic for both the
flat and the hyperbolic metric. This is the case for certain surfaces which are
translation and half-turn tiled by an Euclidean rectangle. We explore them in
this paper. Their multi-geodesic tessellation provides a mechanical way to
reconstruct a Fuchsian group for them; allows to describe their Teichmueller
disk in terms of Fenchel-Nielsen coordinates; allows for an interpretation in
terms of fractional Dehn twists of the natural PSL_2(Z) action on the
PSL_2(R)-orbit of such surfaces. In many cases the tiling by rectangles allows
to recover an equation for the corresponding algebraic curve, providing a
bridge between the algebraic equation and the hyperbolic structure deduced from
the multi-geodesic tessellation; in other words solving the uniformization
problem for such curves. In fact it also gives a scheme to do uniformization
for infinitely many families of curves. We also discuss some number theoretic
aspects.

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The categories of this paper are: solv-int
34
The categories of this paper are: chem-ph
12
The categories of this paper are: nucl-th
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The categories of this paper are: econ
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The categories of this paper are: quant-ph
31
The categories of this paper are: ao-sci
16
The categories of this paper are: stat
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The categories of this paper are: dg-ga
5
The categories of this paper are: gr-qc
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The categories of this paper are: mtrl-th
4
The categories of this paper are: cond-mat
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The categories of this paper are: patt-sol
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The categories of this paper are: nucl-ex
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The categories of this paper are: q-fin
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The categories of this paper are: adap-org
3
The categories of this paper are: physics