[Paper Review] The primitivity index function for a free group, and untangling closed curves on hyperbolic surfaces. With an appendix by Khalid Bou-Rabee

Neha Gupta, Ilya Kapovich|arXiv (Cornell University)|Nov 20, 2014

Geometric and Algebraic Topology43 references3 citations

TL;DR

This paper introduces and analyzes the primitivity index function for free groups, establishing its connection to residual finiteness and hyperbolic surface topology. It proves that the primitivity index grows at least as fast as a function related to the residual finiteness growth, yielding the first nontrivial lower bounds for untangling closed geodesics in finite covers of hyperbolic surfaces.

ABSTRACT

Motivated by the results of Scott and Patel about "untangling" closed geodesics in finite covers of hyperbolic surfaces, we introduce and study primitivity, simplicity and non-filling index functions for finitely generated free groups. We obtain lower bounds for these functions and relate these free group results back to the setting of hyperbolic surfaces. An appendix by Khalid Bou-Rabee connects the primitivity index function to the residual finiteness growth function for $F_N$.

Motivation & Objective

• To define and analyze the primitivity index function for finitely generated free groups, particularly in relation to subgroup separability and primitive elements.
• To establish lower bounds for the primitivity index function, which in turn provide lower bounds for the minimal degree of finite covers required to lift closed geodesics to simple or non-filling curves on hyperbolic surfaces.
• To connect the primitivity index function to the residual finiteness growth function of free groups, leveraging results from geometric group theory.
• To address the quantitative behavior of untangling closed geodesics in finite covers, a problem motivated by Scott and Patel’s results on subgroup separability and curve lifting.
• To provide the first known lower bounds for the function $ f_{ ext{prim}}(n; F_N) $, which governs the minimal cover degree needed to realize a group element as primitive in a finite-index subgroup.

Proposed method

• Define the primitivity index $ d_{ ext{prim}}(g; F_N) $ as the minimal index of a finite-index subgroup $ H rianglelefteq F_N $ in which $ g $ is a primitive element.
• Introduce the primitivity index function $ f_{ ext{prim}}(n; F_N) $ as the maximum of $ d_{ ext{prim}}(g) $ over all nontrivial, non-proper-power elements $ g \in F_N $ of word length $ \leq n $.
• Establish a key inequality: $ \operatorname{RF}_G(n) \leq f_{\text{prim}}(4n+4) $, linking the primitivity index to the residual finiteness growth function $ \operatorname{RF}_G(n) $.
• Apply deep results from Kozma and Thom on residual finiteness growth to derive a lower bound: $ f_{\text{prim}}(4n+4) \geq \exp\left(\left(\frac{\log n}{C\log\log n}\right)^{1/4}\right) $.
• Use commutator constructions $ \gamma_n = [w_n, w_n^a] $ for non-commuting elements $ w_n, a \in F_N $ to ensure $ \gamma_n $ is not a proper power, enabling the primitivity argument.
• Leverage the fact that if $ \gamma_n $ is primitive in $ H $, then $ w_n $ or $ w_n^a $ cannot lie in $ H $, so $ [F_N : H] \geq \operatorname{D}(w_n) = \operatorname{RF}_G(n) $, leading to the main inequality.

Experimental results

Research questions

• RQ1 What is the asymptotic growth rate of the primitivity index function $ f_{\text{prim}}(n; F_N) $ for a free group $ F_N $ of rank $ N \geq 2 $?
• RQ2 How does the primitivity index function relate to the residual finiteness growth function $ \operatorname{RF}_G(n) $ in free groups?
• RQ3 What lower bounds can be established for the minimal degree of a finite cover of a hyperbolic surface that lifts a closed geodesic to a simple or non-filling curve?
• RQ4 Can the primitivity index function be used to quantify the complexity of untangling closed geodesics in finite covers of hyperbolic surfaces?
• RQ5 Do the primitivity index and residual finiteness growth functions in free groups exhibit the same asymptotic behavior?

Key findings

• The paper establishes the first nontrivial lower bound for the primitivity index function: $ f_{\text{prim}}(4n+4) \geq \exp\left(\left(\frac{\log n}{C\log\log n}\right)^{1/4}\right) $ for some constant $ C > 0 $ and sufficiently large $ n $.
• It proves that $ \operatorname{RF}_G(n) \leq f_{\text{prim}}(4n+4) $, showing that the primitivity index function dominates the residual finiteness growth function up to a linear shift in the argument.
• Under Babai’s conjecture on Cayley graph diameters, the lower bound improves to $ f_{\text{prim}}(4n+4) \geq n^{1/(C\log\log n)} $ for large $ n $.
• The results imply that the minimal cover degree required to lift a closed geodesic of length $ L $ to a simple or non-filling curve on a hyperbolic surface grows at least as fast as a function of order $ \exp\left(\left(\frac{\log L}{C\log\log L}\right)^{1/4}\right) $.
• The paper resolves the first known lower bound for the function $ f_{\Sigma}(L) $, which measures the maximal minimal cover degree for lifting closed geodesics of length $ \leq L $.
• The connection between primitivity and residual finiteness reveals that the primitivity index function captures a deeper structural complexity in free groups than previously understood.