Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userCommuting conjugates of finite-order mapping classes
120
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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 for two finite-order mapping classes to have commuting conjugates in $text{Mod}(S_g)$. As an application of this result, we show that any finite-order mapping class, whose corresponding orbifold is not a sphere, has a conjugate that lifts under any finite-sheeted cover of $S_g$. Furthermore, we show that any torsion element in the centralizer of an irreducible finite order mapping class is of order at most $2$. We also obtain conditions for the primitivity of a finite-order mapping class. Finally, we describe a procedure for determining the explicit hyperbolic structures that realize two-generator finite abelian groups of $text{Mod}(S_g)$ as isometry groups.
rate research
Read More
We prove that pseudo-Anosov mapping classes are generic with respect to certain notions of genericity reflecting that we are dealing with mapping classes.
We identify the leading order term of the asymptotic expansion of the Witten-Reshetikhin-Turaev invariants for finite order mapping tori with classical invariants for all simple and simply-connected compact Lie groups. The square root of the Reidemeister torsion is used as a density on the moduli space of flat connections and the leading order term is identified with the integral over this moduli space of this density weighted by a certain phase for each component of the moduli space. We also identify this phase in terms of classical invariants such as Chern-Simons invariants, eta invariants, spectral flow and the rho invariant. As a result, we show agreement with the semiclassical approximation as predicted by the method of stationary phase.
We state Asymptotic Expansion and Growth Rate conjectures for the Witten-Reshetikhin-Turaev invariants of arbitrary framed links in 3-manifolds, and we prove these conjectures for the natural links in mapping tori of finite-order automorphisms of marked surfaces. Our approach is based upon geometric quantisation of the moduli space of parabolic bundles on the surface, which we show coincides with the construction of the Witten-Reshetikhin-Turaev invariants using conformal field theory, as was recently completed by Andersen and Ueno.
Let $Gamma_g$ denote the orientation-preserving Mapping Class Group of the genus $ggeq 1$ closed orientable surface. In this paper we show that for fixed $g$, every finite group occurs as a quotient of a finite index subgroup of $Gamma_g$.
For some $g geq 3$, let $Gamma$ be a finite index subgroup of the mapping class group of a genus $g$ surface (possibly with boundary components and punctures). An old conjecture of Ivanov says that the abelianization of $Gamma$ should be finite. In this note, we prove two theorems supporting this conjecture. For the first, let $T_x$ denote the Dehn twist about a simple closed curve $x$. For some $n geq 1$, we have $T_x^n in Gamma$. We prove that $T_x^n$ is torsion in the abelianization of $Gamma$. Our second result shows that the abelianization of $Gamma$ is finite if $Gamma$ contains a large chunk (in a certain technical sense) of the Johnson kernel, that is, the subgroup of the mapping class group generated by twists about separating curves. This generalizes work of Hain and Boggi.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
