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Michal Stukow

Publications and source records attributed to Michal Stukow.

12 recordsLinked to original sources
arXiv2017

Roots of Dehn twists on nonorientable surfaces

Margalit and Schleimer observed that Dehn twists on orientable surfaces have nontrivial roots. We investigate the problem of roots of a Dehn twist t_c about a nonseparating circle c in the mapping class group M(N_g) of a nonorientable surface N_g of genus g. We explore the existence of roots and, following the work of McCullough, Rajeevsarathy and Monden, give a simple arithmetic description of their conjugacy classes. We also study roots of maximal degree and prove that if we fix an odd integer n>1, then for each sufficiently large g, t_c has a root of degree n in M(N_g). Moreover, for any possible degree n we provide explicit expressions for a particular type of roots of Dehn twists about nonseparating circles in N_g.

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arXiv2016

Subgroups generated by two Dehn twists on a nonorientable surface

Let a and b be two simple closed curves on an orientable surface S such that their geometric intersection number is greater than 1. It is known that the group generated by corresponding Dehn twists t_a and t_b is isomorphic to the free group of rank 2. In this paper we extend this result to the case of a nonorientable surface.

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arXiv2014

A finite presentation for the mapping class group of a nonorientable surface with Dehn twists and one crosscap slide as generators

Let N_{g,s} denote the nonorientable surface of genus g with s boundary components. Recently Paris and Szepietowski obtained an explicit finite presentation for the mapping class group M(N_{g,s}) of the surface N_{g,s}, where s\in{0,1} and g+s>3. Following this work we obtain a finite presentation for the mapping class group M(N_{g,s}) with generators being Dehn twists and one crosscap slide.

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arXiv2007

The twist subgroup of the mapping class group of a nonorientable surface

Let T(N) be the subgroup of the mapping class group of a nonorientable surface N (possibly with punctures and/or boundary components) generated by twists about two-sided circles. We obtain a simple generating set for T(N). As an application we compute the first homology group (abelianization) of T(N).

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arXiv2007

Commensurability of geometric subgroups of mapping class groups

Let M be a surface (possibly nonorientable) with punctures and/or boundary components. The paper is a study of ``geometric subgroups'' of the mapping class group of M, that is subgroups corresponding to inclusions of subsurfaces (possibly disconnected). We characterise the subsurfaces which lead to virtually abelian geometric subgroups. We provide algebraic and geometric conditions under which two geometric subgroups are commensurable. We also describe the commensurator of a geometric subgroup in terms of the stabiliser of the underlying subsurface. Finally, we show some applications of our analysis to the theory of irreducible unitary representations of mapping class groups.

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arXiv2006

Dehn twists on nonorientable surfaces

Let t_a be the Dehn twist about a circle a on an orientable surface. It is well known that for each circle b and an integer n, I(t_a^n(b),b)=|n|I(a,b)^2, where I(,) is the geometric intersection number. We prove a similar formula for circles on nonorientable surfaces. As a corollary we prove some algebraic properties of twists on nonorientable surfaces. We also prove that if M(N) is the mapping class group of a nonorientable surface N, then up to a finite number of exceptions, the centraliser of the subgroup of M(N) generated by the twists is equal to the centre of M(N) and is generated by twists about circles isotopic to boundary components of N.

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arXiv2005

Conjugacy classes of finite subgroups of certain mapping class groups

We give a complete description of conjugacy classes of finite subgroups of the mapping class group of the sphere with r marked points. As a corollary we obtain a description of conjugacy classes of maximal finite subgroups of the hyperelliptic mapping class group. In particular, we prove that for a fixed genus g there are at most five such classes.

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arXiv2005

Small torsion generating sets for hyperelliptic mapping class groups

We prove that both the hyperelliptic mapping class group and the extended hyperelliptic mapping class group are generated by two torsion elements. We also compute the index of the subgroup of the hyperelliptic mapping class group which is generated by involutions and we prove that the extended hyperelliptic mapping class group is generated by three orientation reversing involutions.

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