The Identity Problem in virtually solvable matrix groups over algebraic numbers

TL;DR

This work proves the Identity Problem and Group Problem are decidable for virtually solvable subgroups of $GL(d, \overline{\mathbb{Q}})$, by reducing to a metabelian quotient via a finite-index normal subgroup inside a triangular subgroup. The approach combines automata-theoretic rational subsemigroups with a metabelian embedding $\mathcal{Y} \rtimes \mathbb{Z}^n$, and develops new tools (A-graphs, partial contractions, and position polynomials) to test for identity and group structure. Key contributions include extending nilpotent decidability to groups of finite Prüfer rank and establishing a general decidability result for rational subsemigroups of metabelian groups, which together yield decidability in the virtually solvable setting and move the boundary away from the undecidability frontier given by $F_2 \times F_2$ subgroups. The results open paths to extending decidability to other effectively computable fields and motivate further study of solvable matrix groups beyond $\overline{\mathbb{Q}}$, including potential applications to automata- and polyhedral-based methods in computational group theory.

Abstract

The Tits alternative states that a finitely generated matrix group either contains a nonabelian free subgroup $F_2$, or it is virtually solvable. This paper considers two decision problems in virtually solvable matrix groups: the Identity Problem (does a given finitely generated subsemigroup contain the identity matrix?), and the Group Problem (is a given finitely generated subsemigroup a group?). We show that both problems are decidable in virtually solvable matrix groups over the field of algebraic numbers $\overline{\mathbb{Q}}$. Our proof also extends the decidability result for nilpotent groups by Bodart, Ciobanu, Metcalfe and Shaffrir, and the decidability result for metabelian groups by Dong (STOC'24). Since the Identity Problem and the Group Problem are known to be undecidable in matrix groups containing $F_2 \times F_2$, our result significantly reduces the decidability gap for both decision problems.

Figures (14)

• Figure 1: The disjoint union $\Lambda = \Lambda_1 \sqcup \Lambda_2$ as a subset of $\mathbb{Z}^{s+n}$, where $s = 2, n = 1$.
• Figure 2: An automaton $\mathcal{A}$, where $a_1 = 1$, $a_2 = 1$, $a_3 = -1$, $a_4 = -1$.
• Figure 3: The $\mathcal{A}$-graph $\Gamma(w)$ associated to the accepting run $w = \delta_1 \delta_2 \delta_2 \delta_3 \delta_4$.
• Figure 4: Example of a non face-accessible graph: the strict face $F$ not accessible.
• Figure 5: A non face-accessible graph: $F$ is not accessible. Note that $\Lambda_1, \Lambda_2, \Lambda_3$ are not in the same plane, despite appearing so in the figure.

Theorems & Definitions (107)

• Theorem 1.1
• Lemma 2.0: dong2024semigroup
• Theorem 2.1: beals1999algorithms, ostheimer1999practical
• Theorem 3.1
• Lemma 3.0
• Lemma 3.0
• Theorem 3.1
• Corollary 3.2
• proof
• Lemma 3.2: Composition of kopytov1971solvability and dong2024semigroup