[Paper Review] Geometry and arithmetic of verbal dynamical systems on simple groups

Tatiana Bandman, Fritz Grunewald+2 more|ArXiv.org|Sep 2, 2008

Finite Group Theory Research15 references3 citations

TL;DR

This paper introduces a geometric method using the classical trace map to study periodic points in verbal dynamical systems on simple linear algebraic groups, particularly SL(2,q) and Sz(q). It establishes a connection between group-theoretic properties—such as characterizing finite solvable groups via Engel-like word sequences—and arithmetic dynamics, revealing new phenomena like residual periodicity and proving that the density constant 𝔖_E vanishes for certain elliptic curves.

ABSTRACT

We study dynamical systems arising from word maps on simple groups. We develop a geometric method based on the classical trace map for investigating periodic points of such systems. These results lead to a new approach to the search of Engel-like sequences of words in two variables which characterize finite solvable groups. They also give rise to some new phenomena and concepts in the arithmetic of dynamical systems.

Motivation & Objective

• To understand the geometric and dynamical mechanisms behind recent group-theoretic results characterizing finite solvable groups via two-variable identities.
• To develop a novel geometric method based on the trace map for analyzing periodic points in verbal dynamical systems on algebraic groups.
• To explore the arithmetic-geometric properties of dynamical systems, particularly the concept of residual periodicity, and its implications for local-global behavior.
• To investigate the density constant 𝔖_E in the context of elliptic curves and its vanishing behavior, especially in relation to division fields and Galois representations.
• To generalize the framework to group schemes and residually periodic dynamical systems, suggesting new directions in arithmetic dynamics.

Proposed method

• Utilizes the classical trace map over finite fields to analyze periodic points in dynamical systems defined by word maps on simple groups.
• Applies surjectivity theorems for the trace map to construct and verify dynamical systems with desired periodic behavior.
• Employs Galois-theoretic techniques, including the Chebotarev density theorem, to study the distribution of primes for which certain group-theoretic conditions hold.
• Introduces the concept of residual periodicity, where a dynamical system is periodic modulo every finite quotient but not globally periodic.
• Analyzes the arithmetic of elliptic curves via Galois representations and division fields to determine the vanishing of the density constant 𝔖_E.
• Uses explicit constructions of subgroups of GL₂(ℤ/nℤ) to model possible Galois image structures and test conditions for 𝔖_E > 0.

Experimental results

Research questions

• RQ1 How can geometric methods based on the trace map be used to analyze periodic points in verbal dynamical systems on simple groups?
• RQ2 What is the role of the trace map in explaining the mechanism behind the construction of Engel-like word sequences that characterize finite solvable groups?
• RQ3 Under what conditions does the density constant 𝔖_E vanish, and how is this related to the structure of division fields and Galois representations?
• RQ4 Can the concept of residual periodicity be generalized to other algebraic groups and dynamical systems?
• RQ5 What is the relationship between the vanishing of 𝔖_E and the existence of rational torsion points over certain number fields?

Key findings

• The trace map over finite fields is surjective for SL(2,q), enabling the construction of dynamical systems with controlled periodic behavior.
• For G = SL(2,q), the dynamical system D₁ has a fixed point outside the forbidden set for all q > 3, confirming a key step in characterizing finite solvable groups.
• The density constant 𝔖_E vanishes for elliptic curves E over ℚ where ℚ(√Δ_E) ⊆ ℚ(E[2]) ∩ ℚ(E[3]), as demonstrated by the curve with conductor 30 and discriminant Δ_E = -15.
• For the elliptic curve E with complex multiplication by ℤ[√-15], the set {c_E(p) : p ∤ N_E} = {2,3}, and 𝔖_E = 0, showing that the constant can vanish even when E(ℚ) has infinite order.
• Serre curves satisfy 𝔖_E > 0, and since almost all elliptic curves over ℚ are Serre curves, this suggests that 𝔖_E > 0 is generic in a natural density sense.
• Explicit subgroups G ≤ GL₂(ℤ/3ℓℤ) are constructed where Ω_{3ℓ}(G) = G but Ω_3(G mod 3) ≠ G mod 3 and Ω_ℓ(G mod ℓ) ≠ G mod ℓ, suggesting possible Galois image realizations for curves with 𝔖_E = 0.