Crystallographic groups with trivial center and outer automorphism group

TL;DR

This paper constructs specific crystallographic groups in any dimension greater than or equal to two that have both trivial center and trivial outer automorphism group, advancing understanding of their automorphism structures.

Contribution

It provides explicit examples of crystallographic groups with trivial center and outer automorphism group for all dimensions n ≥ 2, a new construction in geometric group theory.

Findings

Existence of crystallographic groups with trivial center and outer automorphism group in all dimensions n ≥ 2

Explicit construction methods for such groups

Insights into automorphism structures of crystallographic groups

Abstract

Let $\Gamma$ be a crystallographic group of dimension $n,$ i.e. a discrete, cocompact subgroup of $\operatorname{Isom}(\mathbb{R}^n)$ = $O(n)\ltimes\mathbb{R}^n.$ For any $n\geq 2,$ we construct a crystallographic group with a trivial center and a trivial outer automorphism group.