Strongly convergent unitary representations of limit groups

Louder, Larsen; Magee, Michael; Hide, Will

Abstract

We prove that all finitely generated fully residually free groups (limit groups) have a sequence of finite dimensional unitary representations that ‘strongly converge’ to the regular representation of the group. The corresponding statement for finitely generated free groups was proved by Haagerup and Thorbjørnsen in 2005. In fact, we can take the unitary representations to arise from representations of the group by permutation matrices, as was proved for free groups by Bordenave and Collins.

As for Haagerup and Thorbjørnsen, the existence of such representations implies that for any non-abelian limit group, the Ext-invariant of the reduced C⁎-algebra is not a group (has non-invertible elements).

Citation

Louder, L., Magee, M., & Hide, W. (2025). Strongly convergent unitary representations of limit groups. Journal of Functional Analysis, 288(6), Article 110803. https://doi.org/10.1016/j.jfa.2024.110803

• Journal Article Type: Article
• Acceptance Date: Dec 19, 2024
• Online Publication Date: Dec 30, 2024
• Publication Date: Mar 15, 2025
• Deposit Date: Jan 8, 2025
• Publicly Available Date: Jan 8, 2025
• Journal: Journal of Functional Analysis
• Print ISSN: 0022-1236
• Electronic ISSN: 1096-0783
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 288
• Issue: 6
• Article Number: 110803
• DOI: https://doi.org/10.1016/j.jfa.2024.110803
• Public URL: https://durham-repository.worktribe.com/output/3324195