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Transformation groups 2017. Conference dedicated to Prof. Ernest B. Vinberg on the occasion of his 80th birthday
December 15, 2017 12:00, Moscow, Skolkovo Institute of Science and Technology, room 303


Affine groups acting proper and affine crystallographic groups. Mathematical developments arising from Hilbert 18th problem

G. Soifer

University of Bar Ilan, Israel


https://youtu.be/fibnDz-20Qg

Abstract: The study of affine crystallographic groups has a long history which goes back to Hilbert's 18th problem. More precisely Hilbert (essentially) asked if there is only a finite number, up to conjugacy in $\mathrm{Aff}(\mathbb{R}^n)$, of crystallographic groups $\Gamma$ acting isometrically on $\mathbb{R}^n$. In a series of papers Bieberbach showed that this was so. The key result is the following famous theorem of Bieberbach. A crystallographic group $\Gamma$ acting isometrically on the $n$-dimensional Euclidean space $\mathbb R^n$ contains a subgroup of finite index consisting of translations. In particular, such a group $\Gamma$ is virtually abelian, i.e. $\Gamma$ contains an abelian subgroup of finite index. In 1964 Auslander proposed the following conjecture
The Auslander Conjecture. Every crystallographic subgroup $\Gamma$ of $\mathrm{Aff}(\mathbb{R}^n)$ is virtually solvable, i.e. contains a solvable subgroup of finite index.
In 1977 J. Milnor stated the following question:
Question. Does there exist a complete affinely flat manifold $M$ such that $\pi_1(M) $ contains a free group ?
We will explain ideas and methods, recent and old results related to the above problems.

Language: English


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