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 π-graph automatic by the authors, a compelling question is whether F is graph automatic or π-graph automatic for an appropriate language class π. 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.
Funding source: Australian Research Council
Award Identifier / Grant number: FT110100178
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-1105407
Funding source: Simons Foundation
Award Identifier / Grant number: 31736 to Bowdoin College
Funding statement: The first author is supported by Australian Research Council grant FT110100178. The second author acknowledges support from National Science Foundation grant DMS-1105407 and Simons Foundation grant 31736 to Bowdoin College.
The authors wish to thank Sean Cleary, Bob Gilman, Alexei Miasnikov and especially Sharif Younes for many helpful discussions during the writing of this paper.
References
1 D. Berdinsky and B. Khoussainov, On automatic transitive graphs, Developments in Language Theory, Lecture Notes in Comput. Sci. 8633, Lecture Notes in Comput. Sci. (2014), 1β12. 10.1007/978-3-319-09698-8_1Search in Google Scholar
2 K. S. Brown and R. Geoghegan, An infinite-dimensional torsion-free FPF group, Invent. Math. 77 (1984), 2, 367β381. 10.1007/BF01388451Search in Google Scholar
3 J. Burillo, Quasi-isometrically embedded subgroups of Thompson's group F, J. Algebra 212 (1999), 1, 65β78. 10.1006/jabr.1998.7618Search in Google Scholar
4 J. W. Cannon, W. J. Floyd and W. R. Parry, Introductory notes on Richard Thompson's groups, Enseign. Math. (2) 42 (1996), 3β4, 215β256. Search in Google Scholar
5 S. Cleary, S. Hermiller, M. Stein and J. Taback, Tame combing and almost convexity conditions, Math. Z. 269 (2011), 3β4, 879β915. 10.1007/s00209-010-0759-5Search in Google Scholar
6 S. Cleary and J. Taback, Seesaw words in Thompson's group F, Geometric Methods in Group Theory, Contemp. Math. 372, American Mathematical Society, Providence (2005), 147β159. 10.1090/conm/372/06881Search in Google Scholar
7 M. Elder, A context-free and a 1-counter geodesic language for a BaumslagβSolitar group, Theoret. Comput. Sci. 339 (2005), 2β3, 344β371. 10.1016/j.tcs.2005.03.026Search in Google Scholar
8 M. Elder and J. Taback, π-graph automatic groups, J. Algebra 413 (2014), 289β319. 10.1016/j.jalgebra.2014.04.021Search in Google Scholar
9 D. B. A. Epstein, J. W. Cannon, D. F. Holt, S. V. F. Levy, M. S. Paterson and W. P. Thurston, Word Processing in Groups, Jones and Bartlett, Boston, 1992. 10.1201/9781439865699Search in Google Scholar
10 S. A. Greibach, Remarks on blind and partially blind one-way multicounter machines, Theoret. Comput. Sci. 7 (1978), 3, 311β324. 10.1016/0304-3975(78)90020-8Search in Google Scholar
11 V. S. Guba and M. V. Sapir, The Dehn function and a regular set of normal forms for R. Thompson's group F, J. Aust. Math. Soc. Ser. A 62 (1997), 3, 315β328. 10.1017/S1446788700001038Search in Google Scholar
12 O. Kharlampovich, B. Khoussainov and A. Miasnikov, From automatic structures to automatic groups, Groups Geom. Dyn. 8 (2014), 1, 157β198. 10.4171/GGD/221Search in Google Scholar
13 V. Shpilrain and A. Ushakov, Thompson's group and public key cryptography, Applied Cryptography and Network Security, Lecture Notes in Comput. Sci. 3531, Springer, Berlin (2005), 151β163. 10.1007/11496137_11Search in Google Scholar
14 J. Taback and S. Younes, Tree-based language complexity of Thompson's group F, Groups Complex. Cryptol. 7 (2015), 2, 135β152. 10.1515/gcc-2015-0009Search in Google Scholar
Β© 2016 by De Gruyter
Articles in the same Issue
- Frontmatter
- A class of hash functions based on the algebraic eraserβ’
- Generic case complexity of the Graph Isomorphism Problem
- Thompson's group F is 1-counter graph automatic
- The automorphism group of a finitely generated virtually abelian group
- Factoring multi-power RSA moduli with primes sharing least or most significant bits
- Faster Ate pairing computation on Selmer's model of elliptic curves
- A PTIME solution to the restricted conjugacy problem in generalized Heisenberg groups
- Memory-saving computation of the pairing final exponentiation on BN curves
Articles in the same Issue
- Frontmatter
- A class of hash functions based on the algebraic eraserβ’
- Generic case complexity of the Graph Isomorphism Problem
- Thompson's group F is 1-counter graph automatic
- The automorphism group of a finitely generated virtually abelian group
- Factoring multi-power RSA moduli with primes sharing least or most significant bits
- Faster Ate pairing computation on Selmer's model of elliptic curves
- A PTIME solution to the restricted conjugacy problem in generalized Heisenberg groups
- Memory-saving computation of the pairing final exponentiation on BN curves
