[Paper Review] Almost all one-relator groups with at least three generators are residually finite

Mark Sapir, Iva Špakulová|ArXiv.org|Sep 26, 2008

Geometric and Algebraic Topology7 references4 citations

TL;DR

This paper proves that almost all one-relator groups with three or more generators are residually finite, virtually residually (finite p)-group for large primes p, and coherent, by showing they almost surely embed into ascending HNN extensions of free groups. The proof combines combinatorial group theory with probabilistic methods, including Brownian motion results and non-backtracking random walks on free groups, leveraging deep results from geometric group theory and algebraic geometry to establish generic finiteness properties.

ABSTRACT

We prove that with probability tending to 1, a 1-relator group with at least 3 generators and relator of length n is residually finite, virtually residually (finite p)-group for all sufficiently large p, and coherent. The proof uses both combinatorial group theory and non-trivial results about Brownian motions, bridges and excursions in R^k.

Motivation & Objective

• To determine the generic residual finiteness and coherence properties of one-relator groups with at least three generators.
• To establish that such groups almost surely embed into ascending HNN extensions of free groups, a key structural class with known finiteness properties.
• To analyze the asymptotic behavior of random 1-relator groups under three probabilistic models (NR, CR, IC), showing convergence of probabilities to 1.
• To resolve the question of whether residual finiteness holds generically for 1-relator groups, particularly in the 3+ generator case.
• To extend the understanding of the isomorphism problem and structural properties of 1-relator groups using probabilistic and geometric techniques.

Proposed method

• Uses three probabilistic models (NR, CR, IC) to define random 1-relator groups with k ≥ 3 generators and relator length r, analyzing the limit of probabilities as r → ∞.
• Applies results from geometric group theory, including Feighn and Handel’s coherence result for ascending HNN extensions of free groups.
• Employs Borisov and Sapir’s result that such HNN extensions are residually finite and virtually residually (finite p)-group for large primes p.
• Utilizes Cranston, Hsu, and March’s result on the smoothness of the convex hull boundary of Brownian motion paths in Wiener measure to analyze random walk geometry.
• Applies Olshanskii’s congruence extension property for hyperbolic groups and Kapovich-Schupp-Shpilrain’s generic isomorphism problem solvability in 1-relator groups.
• Uses non-backtracking random walks on free groups and defines 'bad' walks (those with no unique projection to a 0-cell), showing their probability tends to zero for k > 2.

Experimental results

Research questions

• RQ1 What is the asymptotic probability that a random one-relator group with k ≥ 3 generators is residually finite?
• RQ2 Can a random one-relator group with k ≥ 3 generators be embedded into an ascending HNN extension of a free group with probability approaching 1?
• RQ3 Does the generic behavior of one-relator groups with k ≥ 3 generators imply coherence and residual finiteness in the limit as relator length increases?
• RQ4 Why does the 2-generator case differ in generic behavior, and can residual finiteness still hold with probability 1 in that case?
• RQ5 Is every HNN extension of the form H(k,i,w) residually finite, and does this imply generic residual finiteness for 2-generator one-relator groups?

Key findings

• With probability tending to 1 as relator length r → ∞, a random k-generator one-relator group (k ≥ 3) embeds into an ascending HNN extension of a finitely generated free group.
• As a consequence, such groups are residually finite and virtually residually (finite p)-group for all sufficiently large primes p, with probability approaching 1.
• The probability that a random k-generator one-relator group (k > 2) is 'bad'—i.e., has no unique projection to a 0-cell in the associated van Kampen diagram—is asymptotically zero.
• The result implies that no additive constant can bound the length increase in relators to preserve non-residual finiteness, contradicting a conjecture in Baumslag-Miller-Traeger.
• For k ≥ 3, the probability of residual finiteness and coherence converges to 1 across all three probabilistic models (NR, CR, IC), due to equivalence of limit probabilities.
• The 2-generator case remains open, though empirical evidence and theoretical work suggest that H(k,i,w) extensions are generic, and their residual finiteness would imply generic residual finiteness in that case.