The Identity Problem in virtually solvable matrix groups over algebraic numbers

2024-04-02

arXiv.org

AI Summary

This paper investigates the identity problem (whether a given finitely generated subsemigroup contains the identity matrix) and the group problem (whether it forms a group) for virtually solvable linear groups over algebraic number fields. It establishes, for the first time, the decidability of both problems for this class of groups. The approach integrates structural theory of algebraic groups, the Tits alternative, recursive decomposition techniques for solvable groups, and effective algorithms over algebraic number fields. Crucially, the results overcome the long-standing undecidability barrier arising from embeddings of (F_2 imes F_2), thereby substantially extending the known decidable domain—from nilpotent groups and solvable groups of derived length at most two—to the broader class of virtually solvable linear groups over algebraic number fields. This work sets a new benchmark in the decision theory of linear groups.

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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 $F_2 imes F_2$, our result significantly reduces the decidability gap for both decision problems.

Problem

• Algebraic Number Theory
• Identity Problem
• Group Problem

Innovation

• Algebraic Number Theory
• Matrix Groups
• Identity Problem