Citation Link: https://doi.org/10.1007/s00224-023-10122-2
Subgroup Membership in GL(2,Z)
Publication Type
Article
Author(s)
Subjects
Algorithm
Artificial intelligence
Computer science
Date Issued
2023
Abstract
It is shown that the subgroup membership problem for a virtually free group can be decided in polynomial time when all group elements are represented by so-called power words, i.e., words of the form $p_1^{z_1} p_2^{z_2} \cdots p_k^{z_k}$. Here the $p_i$ are explicit words over the generating set of the group and all $z_i$ are binary encoded integers. As a corollary, it follows that the subgroup membership problem for the matrix group $\textsf{GL}(2,\mathbb {Z})$ can be decided in polynomial time when elements of $\textsf{GL}(2,\mathbb {Z})$ are represented by matrices with binary encoded integers. For the same input representation, it also shown that one can compute in polynomial time the index of a given finitely generated subgroup of $\textsf{GL}(2,\mathbb {Z})$.
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