The Reidemeister and the Nielsen numbers: growth rate, asymptotic behavior, dynamical zeta functions and the Gauss congruences

This paper investigates the growth rates, asymptotic behaviors, and Gauss congruences of Reidemeister and Nielsen coincidence numbers for endomorphisms of torsion-free nilpotent groups and maps on compact nilmanifolds, while also establishing the rationality of the Nielsen coincidence zeta function.

The Gist

Imagine you are standing in a vast, infinite maze made of twisting corridors and dead ends. This maze represents a mathematical object called a group. Now, imagine you have a magical map (an endomorphism) that tells you how to move through this maze. If you follow the map, you might end up in a different spot, or you might loop back to where you started.

This paper, written by Alexander Fel'shtyn and Mateusz Slomiany, is essentially a study of patterns in chaos. It asks: "If we keep applying this magical map over and over again, how do the patterns of movement behave? Do they grow predictably? Do they follow secret rules?"

Here is a breakdown of their findings using everyday analogies:

1. The Two Main Characters: Reidemeister and Nielsen

To understand the paper, we need to meet the two main "counters" the authors use:

• The Reidemeister Number (The "Total Possibilities" Counter):
  Imagine you are trying to pair up people in a room based on a specific rule. The Reidemeister number counts every single possible way you can pair them up, even if some pairs are just "ghosts" that don't actually exist in reality. It's a theoretical maximum.

  • The Paper's Goal: They want to know how this number explodes as you repeat the process. Does it double every time? Triple? Does it grow like a virus or like a slow-growing plant?

• The Nielsen Number (The "Real Survivors" Counter):
  Now, imagine you shake the room. Some pairings fall apart because they aren't "sturdy" enough. The Nielsen number counts only the essential, unshakeable pairings that survive any amount of shaking (homotopy).

  • The Paper's Goal: They want to know if these "survivors" also grow in a predictable pattern.

2. The "Tame" Maze

The authors focus on a specific type of maze called a nilpotent group. Think of this as a maze with a very strict, orderly structure. It's not a chaotic, random labyrinth; it has layers, like an onion.

• "Tame": This is a fancy word meaning the maze isn't too wild. If you keep applying the map, the number of pairings doesn't shoot up to infinity instantly; it stays manageable. The authors prove that for these orderly mazes, the growth is very predictable.

3. The Growth Rate: The "Speedometer"

One of the main questions is: How fast do these numbers grow?

• The Analogy: Imagine you are watching a bacterial culture. You want to know the "growth rate." Is it growing at 10% an hour? 50%?
• The Discovery: The authors found a "speedometer" for these mathematical mazes. They discovered that the growth rate isn't random. It is determined by the eigenvalues of the map.
  • What are eigenvalues? Think of them as the natural frequencies or the "resonant notes" of the maze. Just as a guitar string vibrates at a specific pitch, the mathematical structure vibrates at specific growth rates. The authors proved that if you know these "notes," you can calculate exactly how fast the numbers will grow.

4. The "Gauss Congruences": The Secret Rhythm

This is perhaps the most magical part of the paper.

• The Analogy: Imagine a drumbeat. If you tap the drum at regular intervals, there's a rhythm. In math, there are ancient rules called congruences (like the famous Gauss congruences) that say: "If you look at the pattern of numbers at specific intervals, they must line up perfectly, like soldiers in a parade."
• The Discovery: The authors proved that the Reidemeister and Nielsen numbers follow these ancient, secret rhythmic rules. Even though the numbers might look huge and messy, if you check them against these "musical rules," they fit perfectly. It's like discovering that a chaotic jazz improvisation actually follows a strict, hidden sheet music score.

5. The Zeta Function: The "Crystal Ball"

The authors use something called a Zeta Function.

• The Analogy: Imagine you have a crystal ball that takes a sequence of numbers (like 2, 4, 8, 16...) and turns it into a single, smooth formula.
• The Discovery: They proved that for these orderly mazes, this crystal ball is rational. This means the formula is simple and clean (like a fraction), not a messy, infinite knot. Because the formula is clean, it guarantees that the "Gauss Rhythms" (congruences) we mentioned earlier must exist.

6. The Big Picture: Why Does This Matter?

You might ask, "Who cares about counting pairings in abstract mazes?"

• Dynamical Systems: This helps us understand how things move and change over time. Whether it's planets orbiting, fluids flowing, or populations growing, these mathematical tools help predict long-term behavior.
• Topology: It connects the shape of space (geometry) with the numbers that describe movement (algebra).
• The "Asymptotic Behavior": The paper tells us that in the long run (as time goes to infinity), the behavior of these systems settles into a predictable pattern. They don't go crazy; they dance to a specific tune determined by the "eigenvalues" (the natural frequencies) of the system.

Summary in One Sentence

This paper proves that in orderly mathematical worlds, the number of ways things can move and coincide follows a strict, predictable growth rate and a hidden rhythmic pattern, much like a complex machine that, despite its size, runs on a simple, elegant set of gears.

Technical Summary

Here is a detailed technical summary of the paper "The Reidemeister and the Nielsen Numbers: Growth Rate, Asymptotic Behavior, Dynamical Zeta Functions and the Gauss Congruences" by Alexander Fel'shtyn and Mateusz Slomiany.

1. Problem Statement

The paper investigates the asymptotic behavior of sequences of Reidemeister numbers ($R$) and Nielsen numbers ($N$) associated with iterations of maps and endomorphisms. Specifically, it addresses:

• The existence and calculation of the growth rate for sequences $\{R(\phi^n)\}$, $\{R(\phi^n, \psi^n)\}$, and $\{N(f^n, g^n)\}$.
• The rationality of the associated dynamical zeta functions (Reidemeister and Nielsen coincidence zeta functions).
• The validity of Gauss congruences for these sequences.
• The asymptotic behavior (limit sets) of these sequences normalized by their growth rates.

The study focuses on tame systems, where the relevant numbers remain finite for all iterations. The primary algebraic setting is a torsion-free nilpotent group $G$ of finite Pr"ufer rank, and the topological setting is a compact nilmanifold.

2. Methodology

The authors employ a hybrid approach combining topological fixed point theory, group theory, and dynamical systems:

• Algebraic Reduction: They utilize the Reidemeister bijection to translate topological problems (fixed/coincidence classes) into algebraic problems involving twisted conjugacy classes in the fundamental group.
• Isolated Lower Central Series: For a torsion-free nilpotent group $G$, they utilize the isolated lower central series (where factors are torsion-free abelian groups $\cong \mathbb{Z}^d$). This allows the decomposition of global Reidemeister numbers into products of Reidemeister numbers of induced endomorphisms on abelian sections (generalizing the Roman'kov formula).
• Profinite Completions and $p$-adic Analysis: The authors analyze endomorphisms via their profinite completions. They decompose the Reidemeister numbers using $p$-adic norms and eigenvalues of induced maps on the divisible hulls of the abelian sections.
• Spectral Analysis: The growth rates are derived from the eigenvalues of the induced linear maps on the rational vector spaces associated with the abelian factors.
• Zeta Functions and Rationality: They define dynamical zeta functions (exponential generating functions of the counts) and use criteria involving algebraic integers and trace formulas to prove rationality.
• Kronecker's Theorem: To analyze asymptotic behavior, they apply Kronecker's theorem on the density of sequences on tori to determine the structure of limit points.

3. Key Contributions and Results

A. Growth Rates of Reidemeister Numbers

• Single Endomorphism ($\phi$): For a tame endomorphism $\phi$ of a torsion-free nilpotent group $G$, the growth rate $R^\infty(\phi) = \lim_{n\to\infty} R(\phi^n)^{1/n}$ exists.
  • Formula: It is given by the product of the maximum of the absolute values of the eigenvalues (in $\mathbb{C}$ and $\mathbb{Q}_p$) of the induced maps on the abelian factors of the isolated lower central series.
  • Connection to Entropy: For finitely generated abelian groups, $R^\infty(\phi) = \exp(h(\hat{\phi}))$, where $h(\hat{\phi})$ is the topological entropy of the dual map on the Pontryagin dual.
• Coincidence Numbers ($\phi, \psi$): For a tame pair $(\phi, \psi)$, the growth rate $R^\infty(\phi, \psi)$ exists and is explicitly calculated via the eigenvalues of $\phi$ and $\psi$ on the abelian factors. The formula involves terms like $\max(|\xi_i|_\infty, |\eta_i|_\infty)$ and $p$-adic contributions.

B. Rationality of Zeta Functions and Gauss Congruences

• Rationality Criterion: The Reidemeister coincidence zeta function $R_{\phi,\psi}(z)$ is rational if and only if the sequence $R(\phi^n, \psi^n)$ can be expressed as a finite sum of powers of algebraic integers (a "trace-like" form).
• Gauss Congruences: The paper proves that if the zeta function is rational, the sequence satisfies the Gauss congruences:
  $$\sum_{d|n} \mu(n/d) R(\phi^d, \psi^d) \equiv 0 \pmod n$$
  This generalizes classical number-theoretic congruences to topological invariants.

C. Nielsen Numbers on Nilmanifolds

• Equality of Invariants: For a tame pair of maps $(f, g)$ on a compact nilmanifold, the Nielsen coincidence number $N(f^n, g^n)$ equals the Reidemeister coincidence number $R(f^n_\#, g^n_\#)$ of the induced homomorphisms on the fundamental group.
• Nielsen Zeta Function: The Nielsen coincidence zeta function $N_{f,g}(z)$ is proven to be rational under the assumption that the eigenvalues of the induced maps satisfy $|\xi_i| \neq |\eta_i|$.
• Growth Rate: The growth rate $N^\infty(f, g)$ exists and is given by an explicit formula involving the eigenvalues of the induced maps on the abelian factors of the fundamental group.
• Congruences: The sequence $\{N(f^n, g^n)\}$ satisfies the Gauss congruences.

D. Asymptotic Behavior

• The paper analyzes the limit set of the normalized sequence $\{R(\phi^k, \psi^k) / \lambda(\phi, \psi)^k\}$, where $\lambda$ is the spectral radius (inverse of the zeta function's radius of convergence).
• Dichotomy: The set of limit points is either:
  1. A finite set (specifically, the limit points of a periodic sequence), occurring when the arguments of the dominant eigenvalues are rational.
  2. An interval, occurring when the arguments of the dominant eigenvalues are linearly independent over $\mathbb{Q}$ (irrational rotation), leading to dense orbits on a torus.

4. Significance

• Unification of Invariants: The work bridges the gap between algebraic invariants (Reidemeister numbers) and topological invariants (Nielsen numbers) in the context of nilmanifolds, showing they share identical asymptotic properties under tameness.
• Explicit Formulas: It provides closed-form formulas for growth rates in terms of eigenvalues, moving beyond qualitative existence results to computable quantities.
• Dynamical Number Theory: The proof of Gauss congruences for these sequences extends classical number theory (Euler/Gauss congruences) to the realm of dynamical systems and fixed point theory, linking them to the rationality of zeta functions.
• Entropy Connection: It solidifies the relationship between the growth of fixed point classes and topological entropy, particularly for maps on nilmanifolds and their duals.
• Asymptotic Classification: The classification of limit sets (finite vs. interval) provides a deeper understanding of the chaotic nature of these sequences, analogous to results for Lefschetz numbers but extended to coincidence theory.

In summary, the paper establishes a comprehensive framework for understanding the long-term behavior of coincidence counts in nilpotent settings, utilizing spectral theory and $p$-adic analysis to derive precise growth rates, rationality properties, and congruence relations.