INFRA-SOLVMANIFOLDS OF Sol14

Title & Authors
INFRA-SOLVMANIFOLDS OF Sol14
LEE, KYUNG BAI; THUONG, SCOTT;

Abstract
The purpose of this paper is to classify all compact manifolds modeled on the 4-dimensional solvable Lie group $\small{Sol_1^4}$, and more generally, the crystallographic groups of $\small{Sol_1^4}$. The maximal compact subgroup of Isom($\small{Sol_1^4}$) is $\small{D_4={\mathbb{Z}}_4{\rtimes}{\mathbb{Z}}_2}$. We shall exhibit an infra-solvmanifold of $\small{Sol_1^4}$ whose holonomy is $\small{D_4}$. This implies that all possible holonomy groups do occur; the trivial group, $\small{{\mathbb{Z}}_2}$ (5 families), $\small{{\mathbb{Z}}_4}$, $\small{{\mathbb{Z}}_2{\times}{\mathbb{Z}}_2}$ (5 families), and $\small{{\mathbb{Z}}_4{\rtimes}{\mathbb{Z}}_2}$ (2 families).
Keywords
solvmanifolds;infra-solvmanifolds;$\small{Sol^3}$;$\small{Sol_1^4}$;Bieberbach theorems;crystallographic groups;
Language
English
Cited by
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