Splittable Lattices in the metabelian solvable Lie group $\mathbb{R}^n\rtimes\mathbb{R}^m$

TL;DR

This work addresses the problem of classifying splittable lattices in the metabelian solvable Lie group $G=\mathbb{R}^n\rtimes_\eta\mathbb{R}^m$ with $\eta(t)=\exp(t\cdot\Delta)$ for diagonal commuting $\Delta_j$. It develops a lattice construction from $G$-compatible data $(\sigma,\rho)$, proves discreteness and cocompactness, and derives a concrete automorphism structure that yields a practical commensurability criterion in terms of $GL_n(\mathbb{Q})$ and $GL_m(\mathbb{Q})$ relations. The paper provides explicit descriptions of lattice equivalence classes and demonstrates the theory with concrete $n=2$ and $n=3$ examples, illustrating the existence and form of splittable lattices in these solvable groups. The results enhance understanding of lattices in solvable Lie groups and have implications for the geometry of the associated solvmanifolds and potential harmonic-analytic applications on these groups.

Abstract

The purpose of this note is describe and classify the splittable lattices in the completely solvable metabelian Lie group (semidirect product of abelian vector groups) $G:=\mathbb{R}^n\rtimes_η\mathbb{R}^m$, where $η$ is the continuous representation of the topological additive abelian group $\mathbb R^m$ in $\mathbb R^n$ given by $η(t_1,\dots, t_m)=\exp(\sum_{j=1}^{m}t_jΔ_j)$ with $(Δ_j)_{1\leq j\leq m}$ is a set of pairwise commuting diagonal matrices in $\mathbb R^{n\times n}$.

Theorems & Definitions (20)

• Proposition 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Definition 2.5
• Remark 2.6
• Definition 3.1
• Remark 3.2
• Theorem 3.3
• proof