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Subgroup Membership in GL(2,Z)

Author Markus Lohrey

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Author Details

Markus Lohrey
  • Universität Siegen, Germany

Funding

  • Lohrey, Markus: Funded by DFG project LO 748/12-1.

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Markus Lohrey. Subgroup Membership in GL(2,Z). In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 51:1-51:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.STACS.2021.51

Abstract

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • free groups
  • virtually free groups
  • subgroup membership
  • matrix groups

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