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High-dimensional knots corresponding to the fractional Fibonacci groups

Andrzej Szczepański, Andreĭ Vesnin (1999)

Fundamenta Mathematicae

We prove that the natural HNN-extensions of the fractional Fibonacci groups are the fundamental groups of high-dimensional knot complements. We also give some characterization and interpretation of these knots. In particular we show that some of them are 2-knots.

Holonomy groups of flat manifolds with the R ∞ property

Rafał Lutowski, Andrzej Szczepański (2013)

Fundamenta Mathematicae

Let M be a flat manifold. We say that M has the R ∞ property if the Reidemeister number R(f) is infinite for every homeomorphism f: M → M. We investigate relations between the holonomy representation ρ of M and the R ∞ property. When the holonomy group of M is solvable we show that if ρ has a unique ℝ-irreducible subrepresentation of odd degree then M has the R ∞ property. This result is related to Conjecture 4.8 in [K. Dekimpe et al., Topol. Methods Nonlinear Anal. 34 (2009)].

Homotopy types of one-dimensional Peano continua

Katsuya Eda (2010)

Fundamenta Mathematicae

Let X and Y be one-dimensional Peano continua. If the fundamental groups of X and Y are isomorphic, then X and Y are homotopy equivalent. Every homomorphism from the fundamental group of X to that of Y is a composition of a homomorphism induced from a continuous map and a base point change isomorphism.

Le groupe d'automorphismes du groupe modulaire

Tambekou Roger Tchangang (1987)

Annales de l'institut Fourier

Le but de cet article est de donner une autre démonstration plus simple du théorème d’Ivanov (Théorème 1) qui assure que le groupe M g * de toutes les difféotopies d’une surface F g orientable et fermée de genre g ≥ 2 est complet. En étudiant l’action d’un automorphisme quelconque du groupe M g * sur les difféotopies d’ordre fini, on montre que les involutions hyperelliptiques sont globalement préservées. Le théorème d’Ivanov est alors une conséquence d’un résultat de Dyer et Grossmann qui affirm que le groupe...

Moduli spaces of local systems and higher Teichmüller theory

Vladimir Fock, Alexander Goncharov (2006)

Publications Mathématiques de l'IHÉS

Let G be a split semisimple algebraic group over Q with trivial center. Let S be a compact oriented surface, with or without boundary. We define positive representations of the fundamental group of S to G(R), construct explicitly all positive representations, and prove that they are faithful, discrete, and positive hyperbolic; the moduli space of positive representations is a topologically trivial open domain in the space of all representations. When S have holes, we defined two moduli spaces closely...

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