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.

翻译：Tits 替代性指出，有限生成的矩阵群要么包含一个非交换自由子群 $F_2$，要么是虚拟可解的。本文研究了虚拟可解矩阵群中的两个判定问题：恒等问题（给定有限生成的子半群是否包含单位矩阵？）和群问题（给定有限生成的子半群是否构成群？）。我们证明，在代数数域 $\overline{\mathbb{Q}}$ 上的虚拟可解矩阵群中，这两个问题都是可判定的。我们的证明还扩展了 Bodart、Ciobanu、Metcalfe 和 Shaffrir 关于幂零群的判定性结果，以及 Dong（STOC'24）关于亚交换群的判定性结果。由于已知在包含 $F_2 \times F_2$ 的矩阵群中恒等问题和群问题是不可判定的，我们的结果显著缩小了这两个判定问题的可判定性差距。