arXiv subjects

Rita Jiménez Rolland

Publications and source records attributed to Rita Jiménez Rolland.

12 recordsLinked to original sources

arXiv2026

The proper geometric dimension of the mapping class groups of non-orientable surfaces with punctures

We study the proper geometric dimension of {\it full} mapping class groups of non-orientable surfaces with punctures. Building on the computation for closed non-orientable surfaces, we prove that the proper geometric dimension agrees with the virtual cohomological dimension in all but a finite collection of low-complexity cases, for which we obtain explicit bounds. Our results provide the non-orientable counterpart of the corresponding computations for mapping class groups of closed and punctured orientable surfaces.

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arXiv2026

Birman-Hilden theory for big mapping class groups

Let $S$ and $X$ be two connected topological surfaces without boundary, and assume that $S$ is either of infinite type or has negative Euler characteristic. In this paper, we prove that if $p:S\rightarrow X$ is a fully ramified branched covering map, then $p$ satisfies the Birman-Hilden property. This generalizes a theorem of Winarski, and the known results in the literature, to the context of surfaces of infinite type and branched covering maps of infinite degree. As an application, we show that the mapping class group (respectively, the braid group on $k$-strands) of a non-orientable surface of infinite type can be realized as a subgroup of the mapping class group (respectively, the braid group on $2k$-strands) of its orientable double cover.

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arXiv2025

The Euler class of infinite-type surface bundles

We study the Euler class of smooth orientable infinite-type surface bundles with a section. For many such surfaces, we show that this cohomology class is nontrivial, and that the behavior of its powers depends on the genus and the type of ends. As an application, we extend Morita's non-lifting theorem to many infinite-type surfaces, including surfaces of infinite genus.

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arXiv2025

A spine for the decorated Teichmüller space of a punctured non-orientable surface

Building on work of Harer \cite{Ha86}, we construct a spine for the decorated Teichmüller space of a non-orientable surface with at least one puncture and negative Euler characteristic. We compute its dimension, and show that the deformation retraction onto this spine is equivariant with respect to the pure mapping class group of the non-orientable surface. As a consequence, we obtain a model for the classifying space for proper actions of the pure mapping class group of a punctured non-orientable surface, which is of minimal dimension in the case there is a single puncture.

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arXiv2024

The proper geometric dimension of the mapping class group of an orientable surface with punctures

We show that the {\it full} mapping class group of any orientable closed surface with punctures admits a cocompact classifying space for proper actions of dimension equal to its virtual cohomological dimension. This was proved for closed orientable surfaces and for {\it pure} mapping class groups by Aramayona and Martínez Pérez. As a consequence of our result we also obtain the proper geometric dimension of {\it full} spherical braid groups.

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arXiv2024

Torsion at the Threshold for Mapping Class Groups

The mapping class group $Γ_g^ 1$ of a closed orientable surface of genus $g \geq 1$ with one marked point can be identified, by the Nielsen action, with a subgroup of the group of orientation preserving homeomorphims of the circle. This inclusion pulls back the powers of the discrete universal Euler class producing classes $\text{E}^n \in H^{2n}(Γ_g^1;\mathbb{Z})$ for all $n\geq 1$. In this paper we study the power $n=g,$ and prove: $\text{E}^g$ is a torsion class which generates a cyclic subgroup of $H^{2g}(Γ_g^1;\mathbb{Z})$ whose order is a positive integer multiple of $4g(2g+1)(2g-1)$.

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arXiv2024

On the dimension of Harer's spine for the decorated Teichmüller space

In \cite{Ha86} Harer explicitly constructed a spine for the decorated Teichmüller space of orientable surfaces with at least one puncture and negative Euler characteristic. In this paper we point out some instances where his computation of the dimension of this spine is off by $1$ and give the correct dimension.

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arXiv2024

On the large scale geometry of big mapping class groups of surfaces with a unique maximal end

Building on the work of K. Mann and K. Rafi, we analyze the large scale geometry of big mapping class groups of surfaces with a unique maximal end. We obtain a complete characterization of those that are globally CB, which does not require the tameness condition. We prove that, for surfaces with a unique maximal end, any locally CB big mapping class group is CB generated, and we give an explicit criterion for determining which big mapping class groups are CB generated. Finally, we give an example of a non-tame surface whose mapping class group is CB generated but is not globally CB.

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arXiv2023

On the virtually cyclic dimension of normally poly-free groups

In this note we give an upper bound for the virtually cyclic dimension of any normally poly-free group in terms of its length. In particular, this implies that virtually even Artin groups of FC-type admit a finite dimensional model for the classifying space with respect to the family of virtually cyclic subgroups.

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arXiv2023

Periodicity of the pure mapping class group of non-orientable surfaces

We show that the pure mapping class group $\mathcal{N}_{g}^{k}$ of a non-orientable closed surface of genus $g\geqslant 2$ with $k\geqslant 1$ marked points has $p$-periodic cohomology for each odd prime $p$ for which $\mathcal{N}_{g}^{k}$ has $p$-torsion. Using the Yagita invariant and the cohomology classes obtained by the representation of subgroups of order $p$, we obtain that the $p$-period is less than or equal to $4$ when $g\geqslant 3$ and $k\geqslant 1$. Moreover, combining the Nielsen realization theorem and a characterization of the $p$-period given in terms of normalizers and centralizers of cyclic subgroups of order $p$, we show that the $p$-period of $\mathcal{N}_{g}^{k}$ is bounded below by $4$, whenever $\mathcal{N}_{g}^{k}$ has $p$-periodic cohomology, $g\geqslant 3$ and $k\geqslant 0$. These results provide partial answers to questions proposed by G. Hope and U. Tillmann.

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arXiv2023

Commensurators of abelian subgroups and the virtually abelian dimension of mapping class groups

Let $\mathrm{Mod}(S)$ be the mapping class group of a compact connected orientable surface $S$, possibly with punctures and boundary components, with negative Euler characteristic. We prove that for any infinite virtually abelian subgroup $H$ of $\mathrm{Mod}(S)$, there is a subgroup $H'$ commensurable with $H$ such that the commensurator of $H$ equals the normalizer of $H'$. As a consequence we give, for each $n \geq 2$, an upper bound for the geometric dimension of $\mathrm{Mod}(S)$ for the family of abelian subgroups of rank bounded by $n$. These results generalize work by Juan-Pineda--Trujillo-Negrete and Nucinkis--Petrosyan for the virtually cyclic case.

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arXiv2015

Representation stability for the pure cactus group

The fundamental group of the real locus of the Deligne-Mumford compactification of the moduli space of rational curves with $n$ marked points, the pure cactus group, resembles the pure braid group in many ways. As it is the case for several "pure braid like" groups, it is known that its cohomology ring is generated by its first cohomology. In this note we survey what the $FI$-module theory developed by Church, Ellenberg and Farb can tell us about those examples. As a consequence we obtain uniform representation stability for the sequence of cohomology groups of the pure cactus group.

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