2018/10/29 by Khalid Bou-Rabee, Bou-Rabee, Khalid, Daniel Studenmund +1
Computer Science · Mathematics · #20E05 #20E26 #20E36 #20F34 #Algebraic Geometry and Number Theory #Computational Geometry and Mesh Generation #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.1810.11909
openalex publication_date 2018/10/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Γ be the fundamental group of a surface of finite type and Comm(Γ) be its abstract commensurator. Then Comm(Γ) contains the solvable Baumslag--Solitar groups ⟨ a ,b : a b a-1 = bn ⟩ for any n > 1. Moreover, the Baumslag--Solitar group ⟨ a ,b : a b2 a-1 = b3 ⟩ has an image in Comm(Γ) that is not residually finite. Our proofs are computer-assisted. Our results also illustrate that finitely-generated subgroups of Comm(Γ) are concrete objects amenable to computational methods. For example, we give a proof that ⟨ a ,b : a b2 a-1 = b3 ⟩ is not residually finite without the use of normal forms of HNN extensions.