What the study found
The authors found an improved polylogarithmic upper bound for the residual finiteness growth of two-step nilpotent groups. They also show that this bound depends only on the complex Mal’cev completion of the group, and that it is exact when the commutator subgroup is one- or two-dimensional.
Why the authors say this matters
The authors note that exact asymptotics for residual finiteness growth are unknown for many groups, including general nilpotent groups. The findings suggest progress on that broader problem by giving a sharper bound in the two-step nilpotent case.
What the researchers tested
The researchers studied residual finiteness growth, a function that measures the size of a finite quotient needed to detect an element of bounded norm in a finitely generated residually finite group. They focused on two-step nilpotent groups and compared their results with bounds known in the literature.
What worked and what didn't
The improved polylogarithmic upper bound worked for all two-step nilpotent groups considered in the paper. The authors also proved exactness in the special cases where the commutator subgroup has dimension one or two. For the general nilpotent setting, the abstract says exact asymptotics are still unknown.
What to keep in mind
The abstract does not describe the proof details or the full range of assumptions beyond finitely generated residually finite groups. It also does not state whether the conjecture for the general case is proved, only that the authors conjecture their exactness result holds more broadly.
Key points
- The paper gives an improved polylogarithmic upper bound for residual finiteness growth in two-step nilpotent groups.
- The bound depends only on the complex Mal’cev completion of the group.
- The bound is exact when the commutator subgroup is one- or two-dimensional.
- The abstract says exact asymptotics remain unknown for many groups, including general nilpotent groups.
- The authors conjecture that the exactness result extends beyond the special low-dimensional cases.
Disclosure
- Research title:
- Exact residual finiteness growth for some two-step nilpotent groups
- Authors:
- Jonas Deré, Joren Matthys
- Institutions:
- University College West Flanders, University College West Flanders
- Publication date:
- 2026-04-22
- OpenAlex record:
- View
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