What the study found
The authors report an improved polylogarithmic upper bound for the residual finiteness growth of two-step nilpotent groups. They also state that this bound depends only on the group’s complex Mal’cev completion, and that it is exact when the commutator subgroup is one- or two-dimensional.
Why the authors say this matters
The study suggests that these bounds help clarify residual finiteness growth, a function that measures how large a finite quotient must be to detect a group element of bounded norm. The authors note that exact asymptotics are still unknown for many group classes, including general nilpotent groups.
What the researchers tested
The paper studies finitely generated residually finite groups and the residual finiteness growth function (operatorname{RF}_{G}colonmathbb{N}tomathbb{N}), which bounds the size of a finite group needed to detect an element of norm at most (r). The authors focus on two-step nilpotent groups and compare their results with previously known estimates.
What worked and what didn't
The improved upper bound works for two-step nilpotent groups, according to the abstract. The authors also prove exactness in the cases where the commutator subgroup has dimension one or two. They conjecture that exactness should hold more generally, but that broader claim is not proved in the abstract.
What to keep in mind
The abstract does not give the explicit form of the new bound. It also does not describe proof details or limitations beyond the stated scope of two-step nilpotent groups and the special one- and two-dimensional commutator subgroup cases.
Key points
- The paper improves a polylogarithmic upper bound for residual finiteness growth in two-step nilpotent groups.
- The bound is said to depend only on the group’s complex Mal’cev completion.
- The authors prove the bound is exact when the commutator subgroup is one- or two-dimensional.
- The abstract says exact asymptotics remain unknown for many group classes, including general nilpotent groups.
- The authors conjecture that exactness may hold more generally, but this is not proved in the abstract.
Disclosure
- Research title:
- Upper bounds improve for residual finiteness growth in two-step nilpotent groups
- Authors:
- Jonas Deré, Joren Matthys
- Institutions:
- University College West Flanders
- Publication date:
- 2026-04-22
- OpenAlex record:
- View
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