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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 $text{Mod}(S_5)$.
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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)$ 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.
Let $G_{m,n,k} = mathbb{Z}_m ltimes_k mathbb{Z}_n$ be the split metacyclic group, where $k$ is a unit modulo $n$. We derive an upper bound for the diameter of $G_{m,n,k}$ using an arithmetic parameter called the textit{weight}, which depends on $n$, $k$, and the order of $k$. As an application, we show how this would determine a bound on the diameter of an arbitrary metacyclic group.
Let $m,n,r$ be positive integers, and let $G=langle arangle: langle brangle cong mathbb{Z}_n: mathbb{Z}_m$ be a split metacyclic group such that $b^{-1}ab=a^r$. We say that $G$ is {em absolutely split with respect to $langle arangle$} provided that for any $xin G$, if $langle xranglecaplangle arangle=1$, then there exists $yin G$ such that $xinlangle yrangle$ and $G=langle arangle: langle yrangle$. In this paper, we give a sufficient and necessary condition for the group $G$ being absolutely split. This generalizes a result of Sanming Zhou and the second author in [arXiv: 1611.06264v1]. We also use this result to investigate the relationship between metacirculants and weak metacirculants. Metacirculants were introduced by Alspach and Parsons in $1982$ and have been a rich source of various topics since then. As a generalization of this classes of graphs, Maruv siv c and v Sparl in 2008 posed the so called weak metacirculants. A graph is called a {em weak metacirculant} if it has a vertex-transitive metacyclic automorphism group. In this paper, it is proved that a weak metacirculant of $2$-power order is a metacirculant if and only if it has a vertex-transitive split metacyclic automorphism group. This provides a partial answer to an open question in the literature.
Let $Gamma(G,S)$ denote the Cayley graph of a group $G$ with respect to a set $S subset G$. In this paper, we analyze the spectral properties of the Cayley graphs $mathcal{T}_{m,n,k} = Gamma(mathbb{Z}_m ltimes_k mathbb{Z}_n, {(pm 1,0),(0,pm 1)})$, where $m,n geq 3$ and $k^m equiv 1 pmod{n}$. We show that the adjacency matrix of $mathcal{T}_{m,n,k}$, upto relabeling, is a block circulant matrix, and we also obtain an explicit description of these blocks. By extending a result due to Walker-Mieghem to Hermitian matrices, we show that $mathcal{T}_{m,n,k}$ is not Ramanujan, when either $m > 8$, or $n geq 400$.
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