arXiv subjects

Yasushi Kasahara

Publications and source records attributed to Yasushi Kasahara.

8 recordsLinked to original sources

arXiv2026

Global fixed point in low-dimensional surface group deformation space

Under the natural action of the pure mapping class group of a surface of genus at least three, we show that any global fixed point in the low-dimensional deformation space of the surface group corresponds to the trivial representation. A key observation is that such a global fixed point gives rise to a linear representation of the pure mapping class group of the corresponding surface with a marked point. Our argument works directly on the deformation space, without assuming the semisimplicity of representations, and yields a short alternative proof of a special case of a theorem of Landesman and Litt with a slight improvement. We also discuss a possible extension of this approach from global fixed points to finite orbits of the mapping class group action.

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arXiv2023

Crossed homomorphisms and low dimensional representations of mapping class groups of surfaces

We continue the study of low dimensional linear representations of mapping class groups of surfaces initiated by Franks--Handel and Korkmaz. We consider $(2g+1)$-dimensional complex linear representations of the pure mapping class groups of compact orientable surfaces of genus $g$. We give a complete classification of such representations for $g \geq 7$ up to conjugation, in terms of certain twisted $1$-cohomology groups of the mapping class groups. A new ingredient is to use the computation of a related twisted $1$-cohomology group by Morita. The classification result implies in particular that there are no irreducible linear representations of dimension $2g+1$ for $g \geq 7$, which marks a contrast with the case $g=2$.

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arXiv2016

Geometric intersection in representations of mapping class groups of surfaces

We show that the detection of geometric intersection in an arbitrary representation of the mapping class group of surface implies the injectivity of that representation up to center, and vice versa. As an application, we discuss the geometric intersection in the Johnson filtration. Also, we further consider the problem of detecting the geometric intersection between separating simple closed curves in a representation.

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arXiv2015

A uniqueness of periodic maps on surfaces

Kulkarni showed that, if g is greater than 3, a periodic map on an oriented surface S_g of genus g with order more than or equal to 4g is uniquely determined by its order, up to conjugation and power. In this paper, we show that, if g is greater than 30, the same phenomenon happens for periodic maps on the surfaces with orders more than 8g/3 and, for any integer N, there is g > N such that there are periodic maps of S_g of order 8g/3 which are not conjugate up to power each other. Moreover, as a byproduct of our argument, we provide a short proof of Wiman's classical theorem: the maximal order of periodic maps of S_g is 4g+2.

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arXiv2014

On visualization of the linearity problem for mapping class groups of surfaces

We derive two types of linearity conditions for mapping class groups of orientable surfaces: one for once-punctured surface, and the other for closed surface, respectively. For the once-punctured case, the condition is described in terms of the action of the mapping class group on the deformation space of linear representations of the fundamental group of the corresponding closed surface. For the closed case, the condition is described in terms of the vector space generated by the isotopy classes of essential simple closed curves on the corresponding surface. The latter condition also describes the linearity for the mapping class group of compact orientable surface with boundary, up to center.

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arXiv2010

Addendum to: "An expansion of the Jones representation of genus 2 and the Torelli group"

We observe that the determinant of the representation provides a little restriction for the structure of the graded quotients $\mathcal{G}_k^Q{F}$ introduced in both [Algebr. Geom. Topol. 1 (2001) 39-55] and [J. Knot Theory Ramifications 13 (2004) 297-306] that any one of them does not contain the trivial 1-dimensional summand.

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arXiv2008

Remarks on the faithfulness of the Jones representations

We consider the linear representations of the mapping class group of an n-punctured 2-sphere constructed by V. F. R. Jones using Iwahori-Hecke algebras of type A. We show that their faithfulness is equivalent to that of certain related Iwahori-Hecke algebra representation of Artin's braid group of n-1 strands. In the case of n=6, we provide a further restriction for the kernel using our previous result, as well as a certain relation to the Burau representation of degree 4.

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arXiv2000

An expansion of the Jones representation of genus 2 and the Torelli group

We study the algebraic property of the representation of the mapping class group of a closed oriented surface of genus 2 constructed by VFR Jones [Annals of Math. 126 (1987) 335-388]. It arises from the Iwahori-Hecke algebra representations of Artin's braid group of 6 strings, and is defined over integral Laurent polynomials Z[t, t^{-1}]. We substitute the parameter t with -e^{h}, and then expand the powers e^h in their Taylor series. This expansion naturally induces a filtration on the Torelli group which is coarser than its lower central series. We present some results on the structure of the associated graded quotients, which include that the second Johnson homomorphism factors through the representation. As an application, we also discuss the relation with the Casson invariant of homology 3-spheres.

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