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Representations of (1,1)-knots

Alessia Cattabriga, Michele Mulazzani (2005)

Fundamenta Mathematicae

We present two different representations of (1,1)-knots and study some connections between them. The first representation is algebraic: every (1,1)-knot is represented by an element of the pure mapping class group of the twice punctured torus PMCG₂(T). Moreover, there is a surjective map from the kernel of the natural homomorphism Ω:PMCG₂(T) → MCG(T) ≅ SL(2,ℤ), which is a free group of rank two, to the class of all (1,1)-knots in a fixed lens space. The second representation is parametric: every...

Representations of a free group of rank two by time-varying Mealy automata

Adam Woryna (2005)

Discussiones Mathematicae - General Algebra and Applications

In the group theory various representations of free groups are used. A representation of a free group of rank two by the so-calledtime-varying Mealy automata over the changing alphabet is given. Two different constructions of such automata are presented.

Representations of PGL ( 2 ) of a local field and harmonic cochains on graph

Paul Broussous (2009)

Annales de la faculté des sciences de Toulouse Mathématiques

We give combinatorial models for non-spherical, generic, smooth, complex representations of the group G = PGL ( 2 , F ) , where F is a non-Archimedean locally compact field. More precisely we carry on studying the graphs ( X ˜ k ) k 0 defined in a previous work. We show that such representations may be obtained as quotients of the cohomology of a graph X ˜ k , for a suitable integer k , or equivalently as subspaces of the space of discrete harmonic cochains on such a graph. Moreover, for supercuspidal representations, these models...

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