[Paper Review] Delzant's T-invariant, one-relator groups and Kolmogorov complexity

Ilya Kapovich, Paul E. Schupp|arXiv (Cornell University)|May 25, 2003

Geometric and Algebraic Topology10 references2 citations

TL;DR

This paper establishes that for 'almost all' one-relator groups, Delzant's T-invariant—measuring the minimal finite presentation size—is asymptotically comparable to the length of the defining relator. Using isomorphism rigidity of generic one-relator groups and Kolmogorov-Chaitin complexity, the authors derive an asymptotic formula for the number of isomorphism classes of k-generator one-relator groups with cyclically reduced relator of length n: $ I_{k,n} \sim \frac{(2k-1)^n}{nk!2^{k+1}} $.

ABSTRACT

We prove that ``almost generically'' for a one-relator group Delzant's $T$-invariant (which measures the smallest size of a finite presentation for a group) is comparable in magnitude with the length of the defining relator. The proof relies on our previous results regarding isomorphism rigidity of generic one-relator groups and on the methods of the theory of Kolmogorov-Chaitin complexity. We also give a precise asymptotic estimate (when $k$ is fixed and $n$ goes to infinity) for the number $I_{k,n}$ of isomorphism classes of $k$-generator one-relator groups with a cyclically reduced defining relator of length $n$: \[ I_{k,n}\sim \frac{(2k-1)^n}{nk!2^{k+1}}. \] Here $f(n)\sim g(n)$ means that $\lim_{n o\infty} f(n)/g(n)=1$.

Motivation & Objective

• To determine the asymptotic growth rate of isomorphism classes of k-generator one-relator groups with cyclically reduced relators of length n.
• To investigate the behavior of Delzant's T-invariant in generic one-relator groups.
• To establish a quantitative relationship between the T-invariant and the length of the defining relator in almost all one-relator groups.
• To apply methods from Kolmogorov-Chaitin complexity to group-theoretic problems in one-relator groups.

Proposed method

• Leverages isomorphism rigidity results for generic one-relator groups to simplify structural analysis.
• Applies techniques from Kolmogorov-Chaitin complexity to bound the number of distinct presentations.
• Uses asymptotic enumeration to estimate the number of isomorphism classes of one-relator groups.
• Derives the asymptotic formula $ I_{k,n} \sim \frac{(2k-1)^n}{nk!2^{k+1}} $ via combinatorial and complexity-theoretic arguments.
• Analyzes the T-invariant by relating it to the minimal presentation size of generic one-relator groups.
• Establishes that the T-invariant is generically comparable in magnitude to the length of the defining relator.

Experimental results

Research questions

• RQ1 What is the asymptotic number of isomorphism classes of k-generator one-relator groups with cyclically reduced relator of length n?
• RQ2 How does Delzant's T-invariant scale relative to the length of the defining relator in generic one-relator groups?
• RQ3 To what extent can Kolmogorov-Chaitin complexity be used to analyze isomorphism types in one-relator groups?
• RQ4 Are there structural constraints on one-relator groups that make their T-invariant predictable in the generic case?
• RQ5 What is the precise asymptotic growth rate of $ I_{k,n} $ as n tends to infinity with k fixed?

Key findings

• The number of isomorphism classes of k-generator one-relator groups with cyclically reduced relator of length n satisfies $ I_{k,n} \sim \frac{(2k-1)^n}{nk!2^{k+1}} $ as n → ∞.
• For 'almost all' one-relator groups, Delzant's T-invariant is asymptotically comparable to the length of the defining relator.
• The asymptotic formula for $ I_{k,n} $ is derived using isomorphism rigidity and Kolmogorov-Chaitin complexity techniques.
• The T-invariant grows proportionally to the relator length in the generic case, indicating minimal presentation complexity is tightly linked to relator length.
• The result confirms a strong structural regularity in the isomorphism types of generic one-relator groups.
• The analysis shows that the number of distinct isomorphism types grows exponentially with n, but is suppressed by a factor of $ nk!2^{k+1} $.