Subgroup Membership in GL(2,Z)

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 p1z1p2z2⋯pkzk\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p_1^{z_1} p_2^{z_2} \cdots p_k^{z_k}$$\end{document}. Here the pi\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p_i$$\end{document} are explicit words over the generating set of the group and all zi\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$z_i$$\end{document} are binary encoded integers. As a corollary, it follows that the subgroup membership problem for the matrix group GL(2,Z)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textsf{GL}(2,\mathbb {Z})$$\end{document} can be decided in polynomial time when elements of GL(2,Z)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textsf{GL}(2,\mathbb {Z})$$\end{document} 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 GL(2,Z)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textsf{GL}(2,\mathbb {Z})$$\end{document}. [ABSTRACT FROM AUTHOR]