Twin groups representations

2025/07/20 by Mohamad N. Nasser, Nasser, Mohamad N. · 1 citation
Mathematics · #Geometric and Algebraic Topology #Algebraic structures and combinatorial models #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.2507.15005

Abstract

We construct two representations of the twin group T_n, n≥ 2, namely η₁: T_n → Aut(\mathbbF_n) and η₂: T_n → GL_n(ℤ[t^{± 1}]), where \mathbbF_n is a free group with n generators and t is indeterminate. We then analyze some characteristics of these two representations, such as irreducibility and faithfulness. Moreover, we prove that both representations can be extended to the virtual twin group VT_n in the 2-local extension way, for n≥ 2, and we find their 2-local extensions. On the other hand, we obtain a different result for the welded twin group WT_n. More deeply, we show that η₁ cannot be extended to WT_n in the 2-local extension way, for n≥ 3, while η₂ can be extended to WT_n in the 2-local extension way, for n≥ 2, and we find its 2-local extensions.

Citations

• Matrix representations of the twisted virtual braid group and its extensions
• Classification of Homogeneous Local Representations of the Singular Braid Monoid
• Cactus groups, twin groups, and right-angled Artin groups
• Structural aspects of twin and pure twin groups

Cited by

• On the classification and irreducibility of 2-local representations of the twin group T_n

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