Overview
This study investigates the properties and realizability of conjugator length functions within the mathematical domain of finitely generated groups. It focuses on characterizing the complete set of functions mapping natural numbers to natural numbers ($\mathbb{N}\rightarrow \mathbb{N}$) that can be instantiated as conjugator length functions, subject to a standard equivalence relation. Furthermore, the research explores the simultaneous realization of such functions in related group structures.
Approach
The work addresses two primary aspects of conjugator length functions. First, it aims to describe all possible functions from $\mathbb{N}$ to $\mathbb{N}$ that can serve as conjugator length functions for finitely generated groups. This characterization is made explicit — 'up to the standard equivalence' — suggesting a classification based on an established mathematical equivalence criterion for these functions.
Second, the study investigates the concurrent realization of multiple conjugator length functions. Specifically, it demonstrates that any pair of increasing functions, $f,g\colon \mathbb N\to \mathbb N$, can both be simultaneously realized as conjugator length functions. This simultaneous realization occurs within a specific context: for finitely generated groups that are 'commensurable'. The concept of commensurability implies a strong structural relationship between the groups, and the source explicitly states that commensurable groups are, in particular, quasi-isometric.
Findings
- The research describes all functions $\mathbb{N}\rightarrow \mathbb{N}$ that are realizable, up to standard equivalence, as conjugator length functions for finitely generated groups.
- It is shown that any two increasing functions $f,g\colon \mathbb N\to \mathbb N$ can be simultaneously realized as conjugator length functions.
- This simultaneous realization is achieved in finitely generated groups that are commensurable, which are also noted to be quasi-isometric.