Symbolické metody v akcích grup na Cantorově prostoru
| Název práce v češtině: | Symbolické metody v akcích grup na Cantorově prostoru |
|---|---|
| Název v anglickém jazyce: | Symbolic methods in group actions on the Cantor space |
| Akademický rok vypsání: | 2024/2025 |
| Typ práce: | disertační práce |
| Jazyk práce: | |
| Ústav: | Katedra matematické analýzy (32-KMA) |
| Vedoucí / školitel: | Mgr. Michal Doucha, Ph.D. |
| Řešitel: |
| Zásady pro vypracování |
| Every continuous action of a countable group on the Cantor space can be shown to be an inverse limit of subshifts defined over the same group. This useful fact is applied e.g. in computing topological full groups and dimension groups of various homeomorphisms on the Cantor space, i.e. actions of the integers, however it is under-researched for more general groups. The student will try to find new applications of this fact focusing, at least in the beginning, on the new research directions presented in [2] by studying the Polish space of all actions of a fixed countable group on the Cantor space using the compact spaces of subshifts over the same group (such spaces are of their own intrinsic interest and were investigated for Z, resp. Z^d in [7], resp. [3]; we suggest a similar study by the student for groups such as free groups or nilpotent groups).
In general, the emphasis will be on actions of finitely generated groups of geometric origin such as hyperbolic groups as well as groups related to automata theory such as automatic groups and automata groups. The student will learn the necessary background from geometric group theory, symbolic dynamics and automata theory e.g. from the literature below. |
| Seznam odborné literatury |
| [1] M. Coornaert and A. Papadopoulos, Symbolic dynamics and hyperbolic groups, vol. 1539 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1993
[2] M. Doucha, Strong topological Rokhlin property, shadowing, and symbolic dynamics of countable groups, to appear in J.Eur. Math. Soc. (JEMS), (2024) [3] S. Gangloff and A. Núnez, The topological structure of isolated points in the space of Z^d-shifts, arXiv preprint2401.17119, (2024). [4] D. F. Holt, S. Rees, and C. E. Röver, Groups, languages and automata, vol. 88 of London Mathematical SocietyStudent Texts, Cambridge University Press, Cambridge, 2017 [5] A. Kechris, Global aspects of ergodic group actions, Mathematical Surveys and Monographs, 160. American Mathematical Society, Providence, RI, 2010. [6] V. Nekrashevych, Groups and topological dynamics, vol. 223 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, [2022]. [7] R. Pavlov and S. Schmieding, On the structure of generic subshifts, Nonlinearity, 36 (2023), pp. 4904–4953 |