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Iterated Monodromy Groups of Captures and Invariant Trees

Student: Shepelevtseva Anastasia

Supervisor: Vladlen Timorin

Faculty: Faculty of Mathematics

Educational Programme: Mathematics and Mathematical Physics (Master)

Final Grade: 8

Year of Graduation: 2017

Captures were defined by B.Wittner and M.Rees as combinatorial operations making post-critically finite hyperbolic polynomials into post-critically finite rational functions. The Iterated Monodromy Group (IMG) provides a complete invariйant for a Thurston equivalence class of a post-critically finite branched covering. The notion was introduced by Volodimir Nekrashevich. It is an example of a self-similar group and it is a homomorphic image of the fundamental group of the sphere minus the post-critical set. The IMG can be represented by an automaton. Knowing the IMGs helps distinguishing Thurston equivalence classes of different captures. This research focuses on how a combinatorial presentation of a branched covering translates into an explicit presentation of its IMG. An invariant graph containing the post-critical set provides a major tool. The main objective of this project is to obtain an explicit algebraic description of the IMGs of captures.

Full text (added May 31, 2017)

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