Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-19T01:57:35.241Z Has data issue: false hasContentIssue false
  • English
  • Français

On the dynamics of extensions of free-abelian times free groups endomorphisms to the completion

Part of: Abelian groups Smooth dynamical systems: general theory Topological dynamics Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  12 August 2022

André Carvalho
André Carvalho*
Affiliation:
Centre of Mathematics, University of Porto, Porto 4169-007, Portugal (andrecruzcarvalho@gmail.com)
Get access Rights & Permissions [Opens in a new window]

Abstract

We obtain conditions of uniform continuity for endomorphisms of free-abelian times free groups for the product metric defined by taking the prefix metric in each component and establish an equivalence between uniform continuity for this metric and the preservation of a coarse-median, a concept recently introduced by Fioravanti. Considering the extension of an endomorphism to the completion, we count the number of orbits for the action of the subgroup of fixed points (respectively periodic) points on the set of infinite fixed (respectively periodic) points. Finally, we study the dynamics of infinite points: for automorphisms and some endomorphisms, defined in a precise way, fitting a classification given by Delgado and Ventura, we prove that every infinite point is either periodic or wandering, which implies that the dynamics is asymptotically periodic.

Keywords

free groupfree-abelian times free groupsautomorphismsendomorphismscoarse-medianfixed pointsinfinite fixed pointsperiodic pointswandering points

MSC classification

Primary: 20E36: Automorphisms of infinite groups 37C25: Fixed points, periodic points, fixed-point index theory
Secondary: 20K25: Direct sums, direct products, etc. 20E05: Free nonabelian groups 37B25: Lyapunov functions and stability; attractors, repellers
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 65 , Issue 3 , August 2022 , pp. 705 - 735
Check for updates
Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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.)

References

Araújo, V. and Silva, P. V., Hölder conditions for endomorphisms of hyperbolic groups, Comm. Algebra 44 (10) (2016), 4483–4503.CrossRefGoogle Scholar
Bestvina, M. and Handel, M., Train tracks and automorphisms of free groups, Ann. Math. 135 (1992), 1–51.CrossRefGoogle Scholar
Bogopolski, O. and Maslakova, O., An algorithm for finding a basis of the fixed point subgroup of an automorphism of a free group, Int. J. Algebra Comput. 26 (1) (2016), 29–67.CrossRefGoogle Scholar
Bowditch, B. H., Coarse median spaces and groups, Pacific J. Math. 261 (1) (2013), 53–93.CrossRefGoogle Scholar
Bowditch, B. H., Median algebras, preprint (2022).Google Scholar
Carvalho, A., On uniformly continuous endomorphisms of hyperbolic groups, J. Algebra 602 (2022),197–223.CrossRefGoogle Scholar
Cassaigne, J. and Silva, P. V., Infinite periodic points of endomorphisms over special confluent rewriting systems, Ann. Inst. Fourier 59 (2) (2009), 769–810.CrossRefGoogle Scholar
Cassaigne, J. and Silva, P. V., Infinite words and confluent rewriting systems: endomorphism extensions, Internat. J. Algebra Comput. 19 (4) (2009), 443–490.CrossRefGoogle Scholar
Chatterji, I., Fernós, T and Iozzi, A., The median class and superrigidity of actions on CAT(0) cube complexes, J. Topol 9 (2) (2016), 349–400.CrossRefGoogle Scholar
Cooper, D., Automorphisms of free groups have finitely generated fixed point sets, J. Algebra 111 (1987), 453–456.CrossRefGoogle Scholar
Dugundji, J., Topology, Reprinting of the 1966 original, Allyn and Bacon Series in Advanced Mathematics (Boston, MA-London-Sydney, Allyn and Bacon, Inc., 1978).Google Scholar
Delgado, J. and Ventura, E., Algorithmic problems for free-abelian times free groups, J. Algebra 263 (1) (2013), 256–283.CrossRefGoogle Scholar
Fioravanti, E., Roller boundaries for median spaces and algebras, Algebr. Geom. Topol. 20 (3) (2020), 1325–1370.CrossRefGoogle Scholar
Fioravanti, E., Coarse-median preserving automorphisms, Geom. Top. (2021). to appear.Google Scholar
Gaboriau, D., Jaeger, A., Levitt, G. and Lustig, M., An index for counting fixed points of automorphisms of free groups, Duke Math. J. 93 (1998), 425–452.CrossRefGoogle Scholar
Gersten, S. M., Fixed points of automorphisms of free groups, Adv. Math. 64 (1987), 51–85.CrossRefGoogle Scholar
Levitt, G. and Lustig, M., Automorphisms of free groups have asymptotically periodic dynamics, J. Reine Angew. Math. 619 (2008), 1–36.CrossRefGoogle Scholar
Matucci, F. and Silva, P. V., Extensions of automorphisms of self-similar groups, J. Group Theory 24 (5) (2021), 857–897.CrossRefGoogle Scholar
Niblo, G. A., Wright, N. and Zhang, J., A four point characterisation for coarse median spaces., Groups Geom. Dyn. 13 (3) (2019), 939–980.CrossRefGoogle Scholar
Paulin, F., Points fixes D'automorphismes de groupes hyperboliques, Ann. Inst. Fourier 39 (1989), 651–662.CrossRefGoogle Scholar
Rodaro, E., Silva, P. V. and Sykiotis, M., Fixed points of endomorphisms of graph groups, J. Group Theory 16 (4) (2013), 573–583.CrossRefGoogle Scholar
Roller, M. A., Poc sets, median algebras and group actions. An extended study of Dunwoody's construction and Sageev's theorem, preprint, (1998).Google Scholar
Silva, P. V., Fixed points of endomorphisms over special confluent rewriting systems, Monatsh. Math. 161 (4) (2010), 417–447.CrossRefGoogle Scholar
Silva, P. V., Fixed points of endomorphisms of virtually free groups, Pacific J. Math. 263 (1) (2013), 207–240.CrossRefGoogle Scholar
Willard, S., General Topology (Addison-Wesley, Reading, MA, 1970).Google Scholar