Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Membership problems in nilpotent groups

Corentin Bodart

TL;DR

The paper investigates Submonoid Membership and Rational Subset Membership in finitely generated nilpotent groups, proving two central reductions. First, SMM(G) is reducible to RatM in smaller subgroups, enabling a construction of a nilpotent group with decidable SMM but undecidable RatM, thereby confirming a Lohrey–Steinberg conjecture in class-2 nilpotent groups. Second, RatM(H3( Z )) is reducible to the Knapsack problem in the same group, yielding decidability of RatM in H3(Z) and, via combination, decidability of SMM in the filiform 3-step nilpotent group. The approach pivots on reducing rational subsets to bounded regular languages and leveraging geometric analysis of the abelianization image π(X) to handle different convex regions (line, plane, half-plane, cone). Collectively, these results deepen understanding of how rational constraints interact with nilpotent structure and provide algorithmic tools, including a bridge to Knapsack, to decide key decision problems in this core class of groups.

Abstract

We study both the Submonoid Membership problem and the Rational Subset Membership problem in finitely generated nilpotent groups. We give two reductions with important applications. First, Submonoid Membership in any nilpotent group can be reduced to Rational Subset Membership in smaller groups. As a corollary, we prove the existence of a group with decidable Submonoid Membership and undecidable Rational Subset Membership, confirming a conjecture of Lohrey and Steinberg. Second, the Rational Subset Membership problem in $H_3(\mathbb Z)$ can be reduced to the Knapsack problem in the same group, and is therefore decidable. Combining both results, we deduce that the filiform $3$-step nilpotent group has decidable Submonoid Membership.

Membership problems in nilpotent groups

TL;DR

The paper investigates Submonoid Membership and Rational Subset Membership in finitely generated nilpotent groups, proving two central reductions. First, SMM(G) is reducible to RatM in smaller subgroups, enabling a construction of a nilpotent group with decidable SMM but undecidable RatM, thereby confirming a Lohrey–Steinberg conjecture in class-2 nilpotent groups. Second, RatM(H3( Z )) is reducible to the Knapsack problem in the same group, yielding decidability of RatM in H3(Z) and, via combination, decidability of SMM in the filiform 3-step nilpotent group. The approach pivots on reducing rational subsets to bounded regular languages and leveraging geometric analysis of the abelianization image π(X) to handle different convex regions (line, plane, half-plane, cone). Collectively, these results deepen understanding of how rational constraints interact with nilpotent structure and provide algorithmic tools, including a bridge to Knapsack, to decide key decision problems in this core class of groups.

Abstract

We study both the Submonoid Membership problem and the Rational Subset Membership problem in finitely generated nilpotent groups. We give two reductions with important applications. First, Submonoid Membership in any nilpotent group can be reduced to Rational Subset Membership in smaller groups. As a corollary, we prove the existence of a group with decidable Submonoid Membership and undecidable Rational Subset Membership, confirming a conjecture of Lohrey and Steinberg. Second, the Rational Subset Membership problem in can be reduced to the Knapsack problem in the same group, and is therefore decidable. Combining both results, we deduce that the filiform -step nilpotent group has decidable Submonoid Membership.
Paper Structure (18 sections, 20 theorems, 66 equations, 4 figures)

This paper contains 18 sections, 20 theorems, 66 equations, 4 figures.

Table of Contents

  1. Background on rational subsets
  2. Definition
  3. Going to subgroups
  4. Virtually abelian groups
  5. Dimension gain for the Submonoid Membership
  6. Background on nilpotent groups
  7. Proof of Theorem \ref{['thm:semi']}
  8. Proof of Theorem \ref{['cor:exe']}
  9. Reduction to bounded regular languages
  10. The Heisenberg group
  11. Reduction to rational submonoids
  12. Main discussion
  13. $\pi(X)$ spans $\{{\bf 0}\}$, a ray or a line
  14. $\pi(X)$ spans the whole plane
  15. $\pi(X)$ spans a half-plane
  16. ...and 3 more sections

Key Result

Corollary 1

The filiform $3$-step nilpotent group (also called "the Engel group") has decidable Submonoid Membership.

Figures (4)

  • Figure 1: Finite state automaton relative to the Knapsack problem
  • Figure 2: A path decomposed as a product of "freeze frames" of the loop-erasure algorithm.
  • Figure 3: A path in the automaton and its decomposition
  • Figure 4: The path corresponding to $h(N_1,0,0;w;0,\ldots,0,P_m)$.

Theorems & Definitions (40)

  • Remark
  • Corollary 1
  • Theorem : See eg. Tits_for_languages
  • Proposition 1.1: Gilman1987GROUPSWA
  • proof
  • Lemma 1.2
  • proof
  • Remark 1.3
  • Theorem 1.4: EILENBERG1969
  • Corollary 1.5
  • ...and 30 more