Subgroup Membership in GL(2,Z)

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 $$p_1^{z_1} p_2^{z_2} \cdots p_k^{z_k}$$ p 1 z 1 p 2 z 2 ⋯ p k z k . Here the $$p_i$$ p i are explicit words over the generating set of the group and all $$z_i$$ z i are binary encoded integers. As a corollary, it follows that the subgroup membership problem for the matrix group $$\textsf{GL}(2,\mathbb {Z})$$ GL ( 2 , Z ) can be decided in polynomial time when elements of $$\textsf{GL}(2,\mathbb {Z})$$ GL ( 2 , 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 $$\textsf{GL}(2,\mathbb {Z})$$ GL ( 2 , Z ) .