It is shown that the subgroup membership problem for a virtually free group can be decided in polynomial time where all group elements are represented by so-called power words, i.e., words of the form p_1^{z_1} p_2^{z_2} ⋯ 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 GL(2,ℤ) can be decided in polynomial time when all matrix entries are given in binary notation.
@InProceedings{lohrey:LIPIcs.STACS.2021.51, author = {Lohrey, Markus}, title = {{Subgroup Membership in GL(2,Z)}}, booktitle = {38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)}, pages = {51:1--51:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-180-1}, ISSN = {1868-8969}, year = {2021}, volume = {187}, editor = {Bl\"{a}ser, Markus and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.51}, URN = {urn:nbn:de:0030-drops-136961}, doi = {10.4230/LIPIcs.STACS.2021.51}, annote = {Keywords: free groups, virtually free groups, subgroup membership, matrix groups} }
Feedback for Dagstuhl Publishing