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Błażej Szepietowski

Publications and source records attributed to Błażej Szepietowski.

10 recordsLinked to original sources
arXiv2026

The pants graph as a combinatorial model for the Teichmüller space of a non-orientable surface

We study the relation between the pants graph of a non-orientable surface and that of its orientable double cover. Given a non-orientable surface $N$ with orientable double cover $π\colon S\to N$, we construct a natural map between pants graphs induced by lifting pants decompositions. We prove that this map defines a quasi-isometric embedding of $\mathrm{Pants}(N)$ into $\mathrm{Pants}(S)$. Using Brock's quasi-isometry between $\mathrm{Pants}(S)$ and Teichmüller space $\mathrm{Teich}(S)$ endowed with the Weil-Petersson metric, together with the identification of the Teichmüller space of $N$ with the fixed-point locus of the deck involution on $\mathrm{Teich}(S)$, we prove that $\mathrm{Pants}(N)$ is quasi-isometric to the Teichmüller space of $N$ equipped with the induced Weil-Petersson metric.

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arXiv2025

Geometric representations of the braid group on a nonorientable surface

We classify homomorphisms from the braid group on $n$ strands to the pure mapping class group of a nonoriantable surface of genus $g$. For $n\ge 14$ and $g\le 2\lfloor{n/2}\rfloor+1$ every such homomorphism is either cyclic, or it maps standard generators of the braid group to either distinct Dehn twists, or distinct crosscap transpositions, possibly multiplied by the same element of the centralizer of the image.

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arXiv2020

Generating the mapping class group of a nonorientable surface by three torsions

We prove that the mapping class group $\mathcal{M}(N_g)$ of a closed nonorientable surface of genus $g$ different than 4 is generated by three torsion elements. Moreover, for every even integer $k\ge 12$ and $g$ of the form $g=pk+2q(k-1)$ or $g=pk+2q(k-1)+1$, where $p,q$ are non-negative integers and $p$ is odd, $\mathcal{M}(N_g)$ is generated by three conjugate elements of order $k$. Analogous results are proved for the subgroup of $\mathcal{M}(N_g)$ generated by Dehn twists.

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arXiv2019

A note on the curve complex of the 3-holed projective plane

Let $S$ be a projective plane with $3$ holes. We prove that there is an exhaustion of the curve complex $\mathcal{C}(S)$ by a sequence of finite rigid sets. As a corollary, we obtain that the group of simplicial automorphisms of $\mathcal{C}(S)$ is isomorphic to the mapping class group $\mathrm{Mod}(S)$. We also prove that $\mathcal{C}(S)$ is quasi-isometric to a simplicial tree.

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arXiv2017

Generating the mapping class group of a nonorientable surface by crosscap transpositions

A crosscap transposition is an element of the mapping class group of a nonorientable surface represented by a homeomorphism supported on a one-holed Klein bottle and swapping two crosscaps. We prove that the mapping class group of a compact nonorientable surface of genus $g\ge 7$ is generated by conjugates of one crosscap transposition. In the case when the surface is either closed or has one boundary component, we give an explicit set of $g+2$ crosscap transpositions generating the mapping class group.

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arXiv2017

On topological classification of finite cyclic actions on bordered surfaces

In [Tohoku Math. J. 62 (2010), 45--53] the second author showed that, except for a few cases, the order $N$ of a cyclic group of self-homeomorphisms of a closed orientable topological surface $S_g$ of genus $g \geq 2$ determines the group up to a topological conjugation, provided that $N\geq 3g$. The first author et al. undertook in [Collect. Math. 67 (2016), 415--429] a more general problem of topological classification of such group actions for $N>2(g-1)$. In [Rev. R. Acad. Cienc. Exactas Fis. Nat., Ser. A. Mat. (RACSAM) 110 (2016), 303--320] we considered the analogous problem for closed non-orientable surfaces, and in [J. Pure Appl. Algebra 220 (2016) 465--481] - the problem of classification of cyclic actions generated by an orientation reversing self-homeomorphism. The present paper, in which we deal with topological classification of actions on bordered surfaces of finite cyclic groups of order $N>p-1$, where $p$ is the algebraic genus of the surface, completes our project of topological classification of "large" cyclic actions on compact surfaces. We apply obtained results to solve the problem of uniqueness of the actions realising the solutions of the so called minimum genus and maximum order problems for bordered surfaces.

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arXiv2016

Automorphisms of the mapping class group of a nonorientable surface

Let $S$ be a nonorientable surface of genus $g\ge 5$ with $n\ge 0$ punctures, and $\Mcg(S)$ its mapping class group. We define the complexity of $S$ to be the maximum rank of a free abelian subgroup of $\Mcg(S)$. Suppose that $S_1$ and $S_2$ are two such surfaces of the same complexity. We prove that every isomorphism $\Mcg(S_1)\to\Mcg(S_2)$ is induced by a diffeomorphism $S_1\to S_2$. This is an analogue of Ivanov's theorem on automorphisms of the mapping class groups of an orientable surface, and also an extension and improvement of the first author's previous result.

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arXiv2007

A presentation for the mapping class group of the closed non-orientable surface of genus 4

Finite presentations for the mapping class group M(F) are known for arbitrary orientable compact surface F. If F is non-orientable, then such presentations are known only when F has genus at most 3 and few boundary components. In this paper we obtain finite presentation for the mapping class group of the closed non-orientable surface of genus 4 from its action on the so called ordered complex of curves.

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arXiv2007

A presentation for the mapping class group of a non-orientable surface from the action on the complex of curves

We study the action of the mapping class group M(F) on the complex of curves of a non-orientable surface F. We obtain, by using a result of K. S. Brown, a presentation for M(F) defined in terms of the mapping class groups of the complementary surfaces of collections of curves, provided that F is not sporadic, i.e. the complex of curves of F is simply connected. We also compute a finite presentation for the mapping class group of each sporadic surface.

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