Hostname: page-component-5d84bcc8dc-zm2vh Total loading time: 0 Render date: 2026-09-07T09:58:26.994Z Has data issue: false hasContentIssue false
  • English
  • Français

COMPUTING CENTRALISERS IN [FINITELY GENERATED FREE]-BY-CYCLIC GROUPS

Part of: Structure and classification of infinite or finite groups Special aspects of infinite or finite groups

Published online by Cambridge University Press:  04 June 2024

ANDRÉ CARVALHO Open the ORCID record for ANDRÉ CARVALHO [Opens in a new window]
ANDRÉ CARVALHO*
Affiliation:
Center for Mathematics and Applications (NOVA Math), NOVA School of Science and Technology, NOVA University Lisbon, 2829–516 Caparica, Portugal
*
e-mail: andrecruzcarvalho@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that centralisers of elements in [finitely generated free]-by-cyclic groups are computable. As a corollary, given two conjugate elements in a [finitely generated free]-by-cyclic group, the set of conjugators can be computed and the conjugacy problem with context-free constraints is decidable. We pose several problems arising naturally from this work.

Keywords

centralisersfree-by-cyclic groupsconjugacy problemautomorphisms

MSC classification

Primary: 20F10: Word problems, other decision problems, connections with logic and automata
Secondary: 20E05: Free nonabelian groups 20E22: Extensions, wreath products, and other compositions

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 111 , Issue 1 , February 2025 , pp. 118 - 127
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

This work is funded by national funds through the FCT–Fundação para a Ciência e a Tecnologia, I.P., under the scope of the projects UIDB/00297/2020 and UIDP/00297/2020 (Center for Mathematics and Applications).

References

Bartholdi, L. and Silva, P. V., ‘Rational subsets of groups’, in: Handbook of Automata Theory (ed. Pin, J.-E.) (EMS Press, Berlin, 2021), Ch. 23.Google Scholar
Baumslag, G., Cannonito, F. B., Robinson, D. J. S. and Segal, D., ‘The algorithmic theory of polycyclic-by-finite groups’, J. Algebra 142 (1991), 118–149.CrossRefGoogle Scholar
Berstel, J., Transductions and Context-free Languages (Teubner, Stuttgart, 1979).CrossRefGoogle Scholar
Bogopolski, O., Martino, A., Maslakova, O. and Ventura, E., ‘The conjugacy problem is solvable in free-by-cyclic groups’, Bull. Lond. Math. Soc. 38 (2006), 787–794.CrossRefGoogle Scholar
Brinkmann, P., ‘Detecting automorphic orbits in free groups’, J. Algebra 324(5) (2010), 1083–1097.CrossRefGoogle Scholar
Carvalho, A., ‘Eventually fixed points of endomorphisms of virtually free groups’, Q. J. Math., to appear.Google Scholar
Carvalho, A., ‘On generalized conjugacy and some related problems’, Comm. Algebra 51(8) (2022), 3528–3542.CrossRefGoogle Scholar
Carvalho, A., ‘Algebraic and context-free subsets of subgroups’, Theor. Comput. Sci. 390 (2023), Article no. 114229.Google Scholar
Ciobanu, L., Evetts, A. and Levine, A., ‘Effective equation solving, constraints and growth in virtually abelian groups’, Preprint, 2023, arXiv:2309.00475.Google Scholar
Franco, N. and González-Meneses, J., ‘Computation of centralizers in braid groups and Garside groups’, Rev. Mat. Iberoam. 19 (2003), 367–384.CrossRefGoogle Scholar
González-Meneses, J. and Ventura, E., ‘Twisted conjugacy in braid groups’, Israel J. Math. 201 (2014), 455–476.CrossRefGoogle Scholar
Gromov, M., ‘Hyperbolic groups’, in: Essays in Group Theory, Mathematical Sciences Research Institute Publications, 8 (ed. Gersten, S.) (Springer, New York, 1987).Google Scholar
Herbst, T., ‘On a subclass of context-free groups’, RAIRO Theor. Inform. Appl. 25 (1991), 255–272.CrossRefGoogle Scholar
Herbst, T., ‘On a kind of Fatou property of context-free groups’, Theor. Comput. Sci. 93 (1992), 327–331.CrossRefGoogle Scholar
Ladra, M. and Silva, P. V., ‘The generalized conjugacy problem for virtually free groups’, Forum Math. 23 (2011), 447–482.CrossRefGoogle Scholar
Logan, A. D., ‘The conjugacy problem for ascending HNN-extensions of free groups’, Preprint, 2022, arXiv:2209.04357.Google Scholar
Miller, C. F. III, ‘Decision problems for groups—survey and reflections’, in: Algorithms and Classification in Combinatorial Group Theory, Berkeley, CA, 1989, Mathematical Sciences Research Institute Publications, 23 (eds. Baumslag, G. and Miller, C. F. III) (Springer, New York, 1992), 1–59.Google Scholar
Ol’shanskii, A. Y. and Sapir, M. V., ‘Groups with small Dehn functions and bipartite chord diagrams’, Geom. Funct. Anal. 16(6) (2006), 1324–1376.CrossRefGoogle Scholar
Silva, P. V., ‘Recognizable subsets of a group: finite extensions and the abelian case’, Bull. Eur. Assoc. Theor. Comput. Sci. EATCS 77 (2002), 195–215.Google Scholar