arXiv subjects

Porfirio L. León Álvarez

Publications and source records attributed to Porfirio L. León Álvarez.

9 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.

math.GT↗

arXiv2026

Asymptotic and cohomological dimension of surface braid groups and poly-surface groups

In this paper, we determine the asymptotic dimension for all surface braid groups -- including those associated with non-orientable and infinite-type surfaces -- as well as for torsion-free poly-finitely generated surface groups. We demonstrate that for both classes, the virtual cohomological dimension and the asymptotic dimension coincide. For poly-finitely generated surface groups and braid groups of finite-type surfaces, our approach establishes that these groups are virtual duality groups in the sense of Bieri-Eckmann. In the case of infinite-type surfaces, the argument is based on the fact that their braid groups are countable and normally poly-free.

math.GR↗

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

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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arXiv2023

Classifying spaces for families of abelian subgroups of braid groups, RAAGs and graphs of abelian groups

Given a group $G$ and an integer $n\geq 0$ we consider the family $\mathcal{F}_n$ of all virtually abelian subgroups of $G$ of rank at most $n$. In this article we prove that for each $n\ge2$ the Bredon cohomology, with respect to the family $\mathcal{F}_n$, of a free abelian group with rank $k > n$ is nontrivial in dimension $k+n$; this answers a question of Corob Cook, Moreno, Nucinkis and Pasini. As an application, we compute the minimal dimension of a classifying space for the family $\mathcal{F}_n$ for braid groups, right-angled Artin groups, and graphs of groups whose vertex groups are infinite finitely generated virtually abelian groups, for all $n\ge2$. The main tools that we use are the Mayer-Vietoris sequence for Bredon cohomology, Bass-Serre theory, and the Lück-Weiermann construction.

math.GR↗

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.

math.GR↗

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.

math.GR↗

arXiv2022

Classifying spaces for the family of virtually abelian subgroups of orientable $3$-manifold groups

For a group $G$, let $\mathcal{F}_{n}$ be the family of all the subgroups of $G$ containing a subgroup isomorphic to $\mathbb{Z}^{r}$ for some $r=0,1,2,\dots ,n$ of finite index. Joecken, Lafont and Sánchez Saldaña computed the $\mathcal{F}_1$-dimension of 3-manifold groups. The goal of this article is to compute the $\mathcal{F}_n$-geometric dimension of 3-manifold groups for all $n\geq 2$.

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