Mapping Class Group Of Genus Research Articles

Finite Quotients of Symplectic Groups VS Mapping Class Groups

We show that the Schur multiplier of Sp(2g,Z/DZ) is Z/2Z, when D is divisible by 4. We give several proofs of this statement, a first one using Deligne’s non-residual finiteness theorem and recent results of Putman, a second one using K-theory arguments based on the work of Barge and Lannes and a third one based on the Weil representations of symplectic groups arising in abelian Chern-Simons theory. We can also retrieve this way Deligne’s non-residual finiteness of the universal central extension ^ Sp(2g,Z) and a sharp result since ^ Sp(2g,Z) surjects on the non-trivial central extension of Sp(2g,Z/DZ) by Z/2Z, when g � 3. We prove then that the essential second homology of finite quotients of symplectic groups over a Dedekind domain of arithmetic type are torsion groups of uniformly bounded size. In contrast, quantum representations produce for every prime p, finite quotients of the mapping class group of genus g � 3 whose essential second homology has p-torsion. We further derive that all central extensions of the mapping class group are residually finite and deduce that mapping class groups have Serre’s property A2 for trivial modules, contrary to symplectic groups. Eventually we compute the module of coinvariants H2(sp2g(2))Sp(2g,Z/2kZ) = Z/2Z.

Generation of finite subgroups of the mapping class group of genus 2 surface by Dehn twists

In this note we give presentations, up to conjugacy, of all finite subgroups of the mapping class group of a closed surface of genus 2, using the Humphries generators. An application to homology representations is given.

On the completion of the mapping class group of genus two

In this paper, we will study the Lie algebra of the prounipotent radical of the relative completion of the mapping class group of genus two. In particular, we will partially determine a minimal presentation of the Lie algebra by determining the generators and bounding the degree of the relations of it.

The asymptotic behavior of the minimal pseudo-ANOSOV dilatations in the hyperelliptic handlebody groups*

We consider the hyperelliptic handlebody group on a closed surface of genus $g$. This is the subgroup of the mapping class group on a closed surface of genus $g$ consisting of isotopy classes of homeomorphisms on the surface that commute with some fixed hyperelliptic involution and that extend to homeomorphisms on the handlebody. We prove that the logarithm of the minimal dilatation (i.e, the minimal entropy) of all pseudo-Anosov elements in the hyperelliptic handlebody group of genus $g$ is comparable to $1/g$. This means that the asymptotic behavior of the minimal pseudo-Anosov dilatation of the subgroup of genus $g$ in question is the same as that of the ambient mapping class group of genus $g$. We also determine finite presentations of the hyperelliptic handlebody groups.

Trace parameters for Teichmüller space of genus 2 surfaces and mapping class group

We obtain a representation of the mapping class group of genus 2 surface in terms of a coordinate system of the Teichmuller space defined by trace functions.

On the rotation angles of a finite subgroup of a mapping class group

Let $G$ be a finite subgroup of the mapping class group of genus $\sigma$, which acts on a compact Riemann surface of genus $\sigma$. In this paper, we introduce a new method to determine the rotation angle of an element $g\in G$ around the fixed points of $g$. Our main result is Theorem 3.2.

Weierstrass points and Morita–Mumford classes on hyperelliptic mapping class groups

We describe the Morita–Mumford classes on the hyperelliptic mapping class group of genus g⩾2, and prove a slightly weakened version of Akita's conjectures in the hyperelliptic case. The proof involves the notion of Weierstrass points.

Meyer's signature cocycle and hyperelliptic fibrations

We show that the cohomology class represented by Meyer's signature cocycle is of order \(2g+1\) in the 2-dimensional cohomology group of the hyperelliptic mapping class group of genus \(g\). By using the \(1\)-cochain cobounding the signature cocycle, we extend the local signature for singular fibers of genus 2 fibrations due to Y. Matsumoto [18] to that for singular fibers of hyperelliptic fibrations of arbitrary genus \(g\) and calculate its values on Lefschetz singular fibers. Finally, we compare our local signature with another local signature which arises from algebraic geometry.

Conjugacy Classes of the Hyperelliptic Mapping Class Group of Genus 2 and 3

We present tables of conjugacy classes of the hyperelliptic mapping class group of genus 2 and 3, and some theorems on the Sp representation, the Jones representation, and Meyer's function.

On the Reducibility and the η-Invariant of Periodic Automorphisms of Genus 2 Surface

We give a characterization for the reducibility of elements of any finite subgroup of the mapping class group of genus 2 surface in terms of the η-invariant of finite order mapping tori.

Homology of hyperelliptic mapping class groups for surfaces

The cohomology of the hyperelliptic mapping class group of genus g with coefficients in any field has been determined by Boedigheimer-Cohen-Peim. We give an alternative, more geometric and elementary way to compute it when the characteristic of the field is greater than g.

The maximal finite subgroup in the mapping class group of genus $5$

The maximal finite subgroup in the mapping class group of genus $5$

The automorphism group of the Klein curve in the mapping class group of genus $3$

The automorphism group of the Klein curve in the mapping class group of genus $3$