Uniformly Almost Flatness and Solubility in Finitely Generated Groups

2026/04/30 by David Guo
#math.GR #math.CO

paper · pdf

Abstract

Tointon and the author conjectured that, for a finitely generated residually finite group, virtual nilpotence is equivalent to the condition that the diameters of its finite coset spaces admit a uniform polynomial lower bound in terms of their sizes. We first verify this conjecture for the class of finitely generated soluble groups. We then prove that this polynomial lower bound condition implies that the group has a finite-index subgroup whose finite quotients are all soluble. An immediate consequence of these two results is the verification of the conjecture for finitely generated linear groups. In addition, we establish the same conclusion for certain finitely generated abelian-by-cyclic groups under the weaker assumption that their finite quotients satisfy this polynomial lower bound condition.

Citations

• Residually finite groups with uniformly almost flat quotients
• On finite groups with polynomial diameter
• On arithmetic properties of solvable Baumslag-Solitar groups
• A functional analysis proof of Gromov's polynomial growth theorem
• Linear groups - Malcev's theorem and Selberg's lemma
• The structure of approximate groups
• A new proof of Gromov's theorem on groups of polynomial growth

Related

• Residually finite groups with uniformly almost flat quotients
• Arbitrarily large finite quotients imply no uniform bound on depth
• Pseudofinite groups with NIP theory and definability in finite simple groups
• Approximating a group by its solvable quotients
• The virtual first Betti number of soluble groups