Let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g\geq 2$. In this paper, we derive necessary and sufficient conditions under which two torsion elements in $\mathrm{Mod}(S_g)$ will have conjugates that generate a finite metacyclic subgroup of $\mathrm{Mod}(S_g)$. This yields a complete solution to the problem of liftability of periodic mapping classes under finite cyclic covers. As applications of the main result, we show that $4g$ is a realizable upper bound on the order of a non-split metacyclic action on $S_g$ and this bound is realized by the action of a dicyclic group. Moreover, we give a complete characterization of the dicyclic subgroups of $\mathrm{Mod}(S_g)$ up to a certain equivalence that we will call weak conjugacy. Furthermore, we show that every periodic mapping class in a non-split metacyclic subgroup of $\mathrm{Mod}(S_g)$ is reducible. We provide necessary and sufficient conditions under which a non-split metacyclic action on $S_g$ factors via a split metacyclic action. Finally, we provide a complete classification of the weak conjugacy classes of the finite non-split metacyclic subgroups of $\mathrm{Mod}(S_{10})$ and $\mathrm{Mod}(S_{11})$.
Let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g\geq 2$. In this paper, we derive necessary and sufficient conditions under which two torsion elements in $\mathrm{Mod}(S_g)$ will have conjugates that generate a finite split non-abelian metacyclic subgroup of $\mathrm{Mod}(S_g)$. As applications of the main result, we give a complete characterization of the finite dihedral and the generalized quaternionic subgroups of $\mathrm{Mod}(S_g)$ up to a certain equivalence that we will call weak conjugacy. Furthermore, we show that any finite-order m…
We obtain representations of the f-based Mapping Class Group of oriented punctured surfaces from an action of mapping classes on Heisenberg homologies of a circle bundle over surface configurations.
The classification of finite group-actions on closed surfaces of small genus is well-known. In the present paper we are interested in the question of which of these group-actions are bounding (extend to a compact 3-manifold with the surface as its unique boundary component, e.g. to a handlebody) or geometrically bounding (extend to a hyperbolic 3-manifold with totally geodesic boundary), concentrating, as a typical case, on large group-actions on surfaces of genus 3.
The present work introduces new perspectives in order to extend finite group actions from surfaces to 3-manifolds. We consider the Schur multiplier associated to a finite group $G$ in terms of principal $G$-bordisms in dimension two, called $G$-cobordisms. We are interested in the question of when a free action of a finite group on a closed oriented surface extends to a non-necessarily free action on a 3-manifold. We show the answer to this question is affirmative for abelian, dihedral, symmetric and alternating groups. As an application of our methods, we show that every non-necessarily free …
We construct the Fukaya category of a surface with genus greater than one and compute its Grothendieck group. We consider here a topological variant, in which we disregard the area form and use instead an admissibility condition borrowed from Heegaard-Floer theory which ensures invariance under isotopy. We also study action of the mapping class group of the surface on its Fukaya category and establish faithfulness.
We provide the first known example of a finite group action on an oriented surface $T$ that is free, orientation-preserving, and does not extend to an arbitrary (in particular, possibly non-free) orientation-preserving action on any compact oriented 3-manifold $N$ with boundary $\partial N = T$. This implies a negative solution to a conjecture of Domínguez and Segovia, as well as Uribe's evenness conjecture for equivariant unitary bordism groups. We more generally provide sufficient conditions that imply infinitely many such group actions on surfaces exist. Intriguingly, any group with such a …
We let the mapping class group $Γ_{g,1}$ of a genus $g$ surface $Σ_{g,1}$ with one boundary component act on the homology $H_*(F_{n}(Σ_{g,1});\mathbb{Q})$ of the $n^{th}$ ordered configuration space of the surface. We prove that the action is non-trivial when restricted to the $(n-1)^{st}$ stage of the Johnson filtration, for all $n\ge 1$ and $g\ge 2$. We deduce an analogous result for closed surfaces.
