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Un hybride du groupe de Thompson F et du groupe de tresses B°°

Abstract : We study a certain monoid specified by a presentation, denoted P, that is a hybrid of the classical presentation of the infinite braid monoid and of the presentation of Thompson’s monoid. To this end, we use several approaches. First, we describe a convergent rewrite system for P, which provides in particular a solution to the word problem, and makes the hybrid monoid reminiscent of Thompson’s monoid. Next, on the shape of the braid monoid, we use the factor reversing method to analyze the divisibility relation, and show in particular that the hybrid monoid admits cancellation and conditional right lcms. Then, we study Garside combinatorics of the hybrid: for every integer n, we introduce an element ∆(n) as the right lcm of the first (n−1) atoms, and one investigates the left divisors of the elements ∆(n), called simple elements. The main results are a counting of the left divisors of ∆(n) and a characterization of the normal forms of simple elements. We conclude with the construction of several representations of the hybrid monoid in various monoids, in particular a representation in a monoid of matrices whose entries are Laurent polynomials, which we conjecture could be faithful.
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Submitted on : Monday, September 10, 2018 - 12:55:07 PM
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Emilie Tesson. Un hybride du groupe de Thompson F et du groupe de tresses B°°. Théorie des groupes [math.GR]. Normandie Université, 2018. Français. ⟨NNT : 2018NORMC212⟩. ⟨tel-01871101⟩

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