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Thompson's group F is 1-counter graph automatic

Thompson's group F is 1-counter graph automatic Abstract It is not known whether Thompson's group F is automatic. With the recent extensions of the notion of an automatic group to graph automatic by Kharlampovich, Khoussainov and Miasnikov and then to 𝒞-graph automatic by the authors, a compelling question is whether F is graph automatic or 𝒞-graph automatic for an appropriate language class 𝒞. The extended definitions allow the use of a symbol alphabet for the normal form language, replacing the dependence on generating set. In this paper we construct a 1-counter graph automatic structure for F based on the standard infinite normal form for group elements. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Groups Complexity Cryptology de Gruyter

Thompson's group F is 1-counter graph automatic

Thompson's group F is 1-counter graph automatic


It is not known whether Thompson's group F is automatic. With the recent extensions of the notion of an automatic group to graph automatic by Kharlampovich, Khoussainov and Miasnikov and then to C-graph automatic by the authors, a compelling question is whether F is graph automatic or C-graph automatic for an appropriate language class C. The extended definitions allow the use of a symbol alphabet for the normal form language, replacing the dependence on generating set. In this paper we construct a -counter graph automatic structure for F based on the standard infinite normal form for group elements. Keywords: Thompson's group F, automatic group, graph automatic group, C-graph automatic group MSC 2010: 20F65, 68Q45 1 Introduction The notion of an automatic group was introduced in the 1990s based on ideas of Thurston, Cannon, Gilman, Epstein and Holt in the hopes of categorizing the fundamental groups of compact 3-manifolds. As not all nilpotent groups proved to be automatic, the definition required expansion to make it more robust. A simultaneous motivation behind this definition was to use the automatic structure to ease computation within these groups. To that end, it was shown that automatic groups are finitely presented, have word problem solvable in quadratic time and possess at most a quadratic Dehn function [9]. In an automatic group there is a natural set of quasigeodesic normal forms which define the structure, and this normal form representative for any group element can always be found in quadratic time. It is not known whether Thompson's group F is automatic. Gubnd Sapir present a regular normal form for elements of F and prove that the Dehn function of F is quadratic in [11]. The word problem in F is solvable in O(n log n) time (see, for example, [13]) and it is shown in [2] that F is of type FP . However, no one has been able to prove that the group is (or is not) automatic. In [6] it is shown that F cannot have a regular combing by...
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Publisher
de Gruyter
Copyright
Copyright © 2016 by the
ISSN
1867-1144
eISSN
1869-6104
DOI
10.1515/gcc-2016-0001
Publisher site
See Article on Publisher Site

Abstract

Abstract It is not known whether Thompson's group F is automatic. With the recent extensions of the notion of an automatic group to graph automatic by Kharlampovich, Khoussainov and Miasnikov and then to 𝒞-graph automatic by the authors, a compelling question is whether F is graph automatic or 𝒞-graph automatic for an appropriate language class 𝒞. The extended definitions allow the use of a symbol alphabet for the normal form language, replacing the dependence on generating set. In this paper we construct a 1-counter graph automatic structure for F based on the standard infinite normal form for group elements.

Journal

Groups Complexity Cryptologyde Gruyter

Published: May 1, 2016

References