Delzant's T-invariant, Kolmogorov complexity and one-relator groups

2003/05/25 by Ilya Kapovich, Kapovich, Ilya, Paul E. Schupp +2
Computer Science · Mathematics · #03D #68Q30 #Computability, Logic, AI Algorithms #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Primary 20F36 #Secondary 20E36 #Topological and Geometric Data Analysis #math.GR #math.GT #msc:03D #msc:20E36 #msc:20F36 #msc:68Q30

paper · pdf · doi:10.48550/arxiv.math/0305353

A revised version, to appear in Comment. Math. Helv

openalex publication_date 2003/05/25 · arxiv created 2005/01/30 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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∼ \frac(2k-1)ⁿnk!2^k+1. Here f(n)∼ g(n) means that lim_n→∞ f(n)/g(n)=1.

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