Amalgamated Products of Groups II: Measures of Random Normal Forms

TL;DR

This paper introduces measures and densities on normal forms in amalgamated free products of groups, analyzing the probabilistic distribution of stable and unstable forms to understand algorithmic performance.

Contribution

It defines atomic measures and asymptotic densities on normal forms in amalgamated free groups, distinguishing between stable and unstable forms and providing probabilistic estimates for each.

Findings

Stable normal forms are more common than unstable ones.

Standard algorithms perform efficiently on stable forms.

Asymptotic densities of normal form strata are characterized.

Abstract

Let $G=\mathop{A\ast B}\limits_C$ be an amalgamated product of finite rank free groups $A$, $B$ and $C$. We introduce atomic measures and corresponding asymptotic densities on a set of normal forms of elements in $G$. We also define two strata of normal forms: the first one consists of regular (or stable) normal forms, and second stratum is formed by singular (or unstable) normal forms. In a series of previous work about classical algorithmic problems, it was shown that standard algorithms work fast on elements of the first stratum and nothing is known about their work on the second stratum. In main theorems A and B of this paper we give probabilistic and asymptotic estimates of these strata.