One variable equations over the lamplighter group

TL;DR

This work resolves the decidability of one-variable equations over the lamplighter group L2 = Z2 wr Z by translating equations into δ-parametric polynomials and reducing a divisibility problem to piecewise periodic instances. It introduces a Magnus-type embedding and a division-by-f automaton to connect algebraic and geometric viewpoints (xt-grid tracing) and proves decidability with a worst-case exponential algorithm, while showing nearly quadratic generic-case complexity. A central contribution is the reduction of δ-parametric divisibility to decidable, finite-search problems via piecewise periodicity and reduction theorems, providing a robust framework for analyzing equations in wreath products. These results advance understanding of algorithmic solvability in metabelian groups and have potential implications for related Diophantine problems and cryptographic considerations.

Abstract

We prove that one variable equations in the lamplighter group $\MZ_2\wr \MZ$ are decidable and describe an algorithm for solving such equations. The algorithm has super-exponential time complexity in the worst case. We also show that, for most equations, decidability can be determined in nearly quadratic time; that is, the problem admits a nearly quadratic-time solution in the generic case.

Figures (9)

• Figure 1: Tracing the word $w= t^{2} a x t^{-1} x^{-2} a$ on the $xt$-grid produces the sets $N_w = \{ x^0 t^0, x^{1} t^{1} \}$ and $D_w = \{ x^{1}t^0, x^{2}t^0, x^{2}t^{1} \}$ for the polynomials $\mathop{\mathrm{num}}\nolimits(w)$ and $\mathop{\mathrm{den}}\nolimits(w)$. The left and right diagrams show $N_w$ (red points) and $D_w$ (green points) resp.
• Figure 2: Substituting $\delta \leftarrow 1$ to the example in Figure \ref{['fig:tracing_example']}, we can visualize the numerator (see left diagram) and denominator (see right diagram) polynomials on $t$-axis. The sets $N_w$ and $D_w$ are $N_w = \{ z^0, z^2 \}$ and $D_w =\{ z^1, z^2, z^3 \}$.
• Figure 3: Tracing the word $w = t^{1-n} x^{n-1} a t^{-1} x^{-n} a$ creates a path from $(0,0)$ to $(1,n)$ and produces the sets $N_w =\{ x^n t^1, x^0 t^0 \}$ and $D_w =\{ x^1 t^0, x^2 t^0 \dots, x^n t^0, x^2 t^1, x^3 t^1, \dots , x^n t^1 \}$.
• Figure 4: $\Gamma_f$ for $f=z^3+z+1$.
• Figure 5: $\Gamma_f$ for $f=z^2+1$ and its normalization which is the Cayley graph of $D_4\simeq {\mathbb{Z}}_4\rtimes {\mathbb{Z}}_2$ with $P_f=8$.

Theorems & Definitions (76)

• Lemma 2.1
• Theorem 2.2: GathenGerhard2003
• Theorem 2.3: GathenGerhard2003
• Theorem 2.4: GathenGerhard2003
• Proposition 2.5
• proof
• Lemma 3.1
• proof
• Lemma 4.1
• proof