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Register a new userGeometric realizations of cyclic actions on surfaces - II
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Let $ text{Mod}(S_g)$ denote the mapping class group of the closed orientable surface $S_g$ of genus $ggeq 2$. Given a finite subgroup $H leq text{Mod}(S_g)$, let $text{Fix}(H)$ denote the set of fixed points induced by the action of $H$ on the Teichm{u}ller space $text{Teich}(S_g)$. The Nielsen realization problem, which was answered in the affirmative by S. Kerckhoff, asks whether $text{Fix}(H) eq emptyset$, for any given $H$. In this paper, we give an explicit description of $text{Fix}(H)$, when $H$ is cyclic. As consequences of our main result, we provide alternative proofs for two well known results, namely a result of Harvey on $text{dim}(text{Fix}(H))$, and a result of Gilman that characterizes irreducible finite order actions. Finally, we derive a correlation between the orders of irreducible cyclic actions and the filling systems on surfaces.
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Let $ text{Mod}(S_g)$ denote the mapping class group of the closed orientable surface $S_g$ of genus $ggeq 2$, and let $fin text{Mod}(S_g)$ 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 $Psi: text{Mod}(S_g) to text{Sp}(2g; mathbb{Z})$.
Let $text{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $ggeq 2$. In this paper, we derive necessary and sufficient conditions under which two torsion elements in $text{Mod}(S_g)$ will have conjugates that generate a finite split nonabelian metacyclic subgroup of $text{Mod}(S_g)$. As applications of the main result, we give a complete characterization of the finite dihedral and the generalized quaternionic subgroups of $text{Mod}(S_g)$ up to a certain equivalence that we will call weak conjugacy. Furthermore, we show that any finite-order mapping class whose corresponding orbifold is a sphere, has a conjugate that lifts under certain finite-sheeted regular cyclic covers of $S_g$. Moreover, for $g geq 5$, we show the existence of an infinite dihedral subgroup of $text{Mod}(S_g)$ that is generated by the hyperelliptic involution and a root of a bounding pair map of degree $3$. Finally, we provide a complete classification of the weak conjugacy classes of the non-abelian finite split metacyclic subgroups of $text{Mod}(S_3)$ and $text{Mod}(S_5)$.
This note is about a type of quantitative density of closed geodesics on closed hyperbolic surfaces. The main results are upper bounds on the length of the shortest closed geodesic that $varepsilon$-fills the surface.
In this paper we study smooth orientation-preserving free actions of the cyclic group $mathbb Z/m$ on a class of $(n-1)$-connected $2n$-manifolds, $sharp g (S^n times S^n)sharp Sigma$, where $Sigma$ is a homotopy $2n$-sphere. When $n=2$ we obtain a classification up to topological conjugation. When $n=3$ we obtain a classification up to smooth conjugation. When $n ge 4$ we obtain a classification up to smooth conjugation when the prime factors of $m$ are larger than a constant $C(n)$.
We realize Stasheffs multiplihedron geometrically as the moduli space of stable quilted disks. This generalizes the geometric realization of the associahedron as the moduli space of stable disks. We show that this moduli space is the non-negative real part of a complex moduli space of stable scaled marked curves.
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