Session S23 - Group actions in Differential Geometry

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Classification of 6-dimensional almost abelian flat solvmanifolds

Alejandro Tolcachier

FAMAF-UNC, Argentina   -   aletolcachier@gmail.com

Solvmanifolds, i.e. compact manifolds obtained as quotients of simply-connected solvable Lie groups by discrete subgroups (called lattices), are an important class of manifolds. Some of these solvmanifolds admit a flat Riemannian metric induced by a flat left invariant metric on the associated Lie group. Flat solvmanifolds are a particular class of compact flat manifolds. Such a solvmanifold is isometric to a compact quotient $\mathbb{R}^n/\Gamma$ for some discrete subgroup $\Gamma$ of the isometries of $\mathbb{R}^n$ and its fundamental group is isomorphic to $\Gamma$. These groups were characterized by the so called three Bieberbach Theorems and consequently they are called Bieberbach groups.

In general, it is difficult to determine whether a solvable Lie group admits lattices or not, which makes difficult the construction of solvmanifolds. This poster will talk about a special class of flat solvmanifolds arisen from almost abelian flat Lie groups, that is, groups of the form $\mathbb{R}\ltimes_{\phi} \mathbb{R}^d$ for certain action $\phi$. An advantage is that for almost abelian Lie groups there is a criterion to determine all its lattices. We will focus in the classification of 6-dimensional almost abelian flat solvmanifolds. In order to do so, we will have to solve the problem of finding the conjugacy classes of certain matrices in $\mathsf{GL}(5,\mathbb{Z})$ with finite order.

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