Virtual First Betti Number of GGS Groups

2025/05/29 by Andrew Keong Ng, Ng, Andrew
Mathematics · #Geometric and Algebraic Topology #Finite Group Theory Research #Advanced Operator Algebra Research

paper · pdf · doi:10.48550/arxiv.2505.23269

Abstract

We observe a criterion for groups to have vanishing virtual first Betti number and use it to give infinitely many examples of torsion-free, finitely generated, residually finite groups which aren't virtually diffuse. This answers a question raised by Kionke and Raimbault.

Citations

• Group-graded rings satisfying the strong rank condition
• Amenability of groups is characterized by Myhill's Theorem
• Amenable groups with a locally invariant order are locally indicable
• GGS-groups: order of congruence quotients and Hausdorff dimension

Related

• The virtual first Betti number of soluble groups
• Approximating the first L²-betti number of residually finite groups
• Some non-amenable groups
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