Independence of the conjugacy problem and conjugacy separability
Lukas Vandeputte
TL;DR
The paper addresses whether a fast conjugacy problem solution can coexist with low conjugacy separability. It constructs a family $\mathcal{G}$ of recursively-presented, finitely generated, conjugacy separable groups with quotient enumeration such that each $G\in\mathcal{G}$ has a polynomial-time conjugacy problem, yet the conjugacy separability function $\mathrm{Conj}_{G}$ can be made arbitrarily large by choosing an appropriate $d:\mathbb{N}\to\mathbb{N}$. By parameterizing with a computable $d$ that is non-decreasing and satisfies certain fast-compute/fast-compare properties, the authors obtain large $\mathrm{Conj}_{G}$ while preserving efficient solvability of the conjugacy problem. The construction uses central quotients $G_d=G_0/N_d$ and finite quotients $G_{d,I}$, analyzes commutators via the central subgroup $C$, and employs a Malcev completion $\mathbb{Q}\frac{D}{C}$ to derive fractional conjugators, ultimately showing that McKinsey-type algorithms can be arbitrarily far from optimal. This demonstrates a fundamental decoupling between the conjugacy problem and conjugacy separability, with a structured method to control both quotient enumeration and complexity.
Abstract
We construct a class of finitely generated groups which have arbitrarily large conjugacy separability function, but in which the conjugacy problem can be solved in polynomial time, demonstrating that the McKinsey algorithm for the conjugacy problem can have complexity which lies arbitrarily far from the optimum.