Bieberbach groups and fibering flat manifolds of diagonal type

Abstract

[Text taken from Chapter 1]

Hilbert’s problems are twenty-three problems in mathematics published by German mathematician David Hilbert in 1900. Hilbert’s eighteenth problem is related to crystallographic groups. We denote Isom(Rn) to be the group of all isometries of the n-dimensional Euclidean space Rn. One part of the Hilbert’s eighteenth problem is to show that there are only finitely many types of subgroups of Isom(Rn) with compact fundamental domain. This part of the question is solved by L. Bieberbach in 1910. Those subgroups of Isom(Rn)
with compact fundamental domain are now called crystallographic groups. In the thesis, we will study serval properties of torsion-free crystallographic groups.
In Chapter 2, we first introduce the definition of crystallographic groups. We say Γ is an n-dimensional crystallographic group if it is a cocompact and discrete subgroup of Isom(Rn) ∼= O(n)nRn. We say Γ is an n-dimensional Bieberbach group if Γ is a torsion-free n-dimensional crystallographic group. Next, we will present the first Bieberbach theorem. [...]

In Chapter 2, we first introduce the definition of crystallographic groups. [...]

In Chapter 3, we focus on the below conjecture.
Conjecture 1.0.1 (Dekimpe-Penninckx). Let Γ be an n-dimensional Bieberbach group. Then the minimum number of generators of Γ is less than or equal to n.
[...]

In Chapter 4, we will study about an invariant called the diagonal Vasquez invariant. [...]

In Chapter 5, we will discuss diffuseness property of Bieberbach groups. [...]

Results from Chapter 4 and Chapter 5 is a preprint: see Fibering flat manifolds of diagonal type and their fundamental groups arXiv:2011.07381 [math.GR] )

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ORCID for Nansen Petrosyan: orcid.org/0000-0002-2768-5279

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