Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Geometric realizations of cyclic actions on surfaces   Let $ \text{Mod}(S_g)$ denote the mapping class group of the closed
orientable surface $S_g$ of genus $g\geq 2$, and let $f\in \text{Mod}(S_g)$ be
of finite order. We give an inductive procedure to construct an explicit
hyperbolic structure on $S_g$ that realizes $f$ as an isometry. In other words,
this procedure yields an explicit solution to the Nielsen realization problem
for cyclic subgroups of $ \text{Mod}(S_g)$. Furthermore, we give a purely
combinatorial perspective by showing how certain finite order mapping classes
can be viewed as fat graph automorphisms. As an application of our
realizations, we determine the sizes of maximal reduction systems for certain
finite order mapping classes. Moreover, we describe a method to compute the
image of finite order mapping classes and the roots of Dehn twists, under the
symplectic representation $\Psi: \text{Mod}(S_g) \to \text{Sp}(2g;
\mathbb{Z})$.

9
The categories of this paper are: nlin
32
The categories of this paper are: atom-ph
22
The categories of this paper are: chao-dyn
19
The categories of this paper are: eess
24
The categories of this paper are: dg-ga
31
The categories of this paper are: ao-sci
13
The categories of this paper are: quant-ph
33
The categories of this paper are: plasm-ph
0
The categories of this paper are: hep-ph
12
The categories of this paper are: nucl-th
5
The categories of this paper are: gr-qc
35
The categories of this paper are: mtrl-th
17
The categories of this paper are: q-fin
14
The categories of this paper are: math-ph
10
The categories of this paper are: q-bio
28
The categories of this paper are: alg-geom