Residual finiteness properties of some of Halls groups

2025/08/22 by Vandeputte, Lukas
#20F10 #20F69 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2508.16452

Abstract

In this article we study a class of central extensions of ℤ\wrℤ, as first described by Hall. On the one hand, we consider groups of this type with cyclic centre, our construction yields a rich class of groups. In particular we obtain a group that is conjugacy separable with solvable word problem but unsolvable conjugacy problem, we obtain a group with large conjugacy separability growth but small conjugator length function and residual finiteness growth, and we also obtain a class of groups that for most functions f:ℕ→ℕ larger then n³, contain a group G such that the conjugator length of G is given by f. On the other hand we also consider a different group with larger centre. This is the first example of a group where the residual finiteness growth is faster than any polynomial but slower than any exponential.

Citations

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Related

• Arbitrary residual finiteness and conjugacy separability growth
• Full residual finiteness growths of nilpotent groups
• Arbitrarily Large Residual Finiteness Growth
• A robust family of residually finite groups; spectra of residual finiteness growth, computability properties, and other applications (with an appendix by Arman Darbinyan and Emmanuel Rauzy)
• Groups with fast-growing conjugator length functions