Porfirio Leandro Leon Alvarez: Virtually Abelian Dimension for 3-Manifold Groups

Porfirio Leandro Leon Alvarez, Instituto de Matematicas, UNAM
Title: Virtually Abelian Dimension for 3-Manifold Groups
Given a group $\Gamma$, we say a collection $\mc F$ of subgroups of $\Gamma$ is a family if it is non-empty, closed under conjugation and under taking subgroups. Fixing a group $\Gamma$ and a family $\mc F$ of subgroups of $\Gamma$, we say that a $\Gamma$-CW-complex $X$ is a model for the classifying space $E_{\mc F}\Gamma$ if every isotropy group of $X$ belongs to the family $\mc F$ and the fixed point set $X^H$ is contractible whenever $H$ belongs to $\mc F$. It can be shown that a model for the classifying space $E_{\mc F}\Gamma$ always exists and it is unique up to $\Gamma$-homotopy equivalence.

We define the $\mc F$-geometric dimension of $\Gamma$, denoted as $gd_{\mc F}(\Gamma)$, as the minimal dimension of the models for the classifying space $E_{\mc F}\Gamma$.

Now, let $\Gamma$ be the fundamental group of a 3-manifold. Define the family $\mc F_n$ as the family of virtually $\Z^r$ subgroups for $0 \leq r \leq n$.
In joint work with Luis Jorge S{\'a}nchez Salda{\~n}a we computed $gd_{\mc F_n}(\Gamma)$ for all $n\geq 2$. In this talk I will give an explicit formula for this dimension.\\
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