Proper CAT(0) actions of unipotent-free linear groups

2021/07/20 by Sami Douba, Douba, Sami
Mathematics · #20F65 (Secondary) #20F67 (Primary) #Advanced Algebra and Geometry #Advanced Operator Algebra Research #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.2107.09490

openalex publication_date 2021/07/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Γ be a finitely generated group of matrices over ℂ. We construct an isometric action of Γ on a complete CAT(0) space X such that the restriction of this action to any subgroup of Γ containing no nontrivial unipotent elements is well behaved. As an application, we show that if M is a graph manifold that does not admit a nonpositively curved Riemannian metric, then any finite-dimensional ℂ-linear representation of π₁(M) maps a nontrivial element of π₁(M) to a unipotent matrix. In particular, the fundamental groups of such 3-manifolds do not admit any faithful finite-dimensional unitary representations.

Related

• Aspects of non positive curvature for linear groups with no infinite order unipotents
• Finiteness of CAT(0) group actions
• Rigidity of matrix group actions on CAT(0) spaces with possible parabolic isometries and uniquely arcwise connected spaces
• The rhombic dodecahedron and semisimple actions of Aut(F_n) on CAT(0) spaces
• Properties of linear groups with restricted unipotent elements