The algebra of strand splitting. I. A braided version of Thompson's group V

By Matthew G. Brin. 2004

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We construct a braided version BV of Thompson’s group V that surjects onto V . The group V is the third of three well known groups F , T and V created by Thompson in the 1960s that have been heavily studied since. See [6] and Section 4 of [5] for an introduction to Thompson’s groups. The group V is a subgroup of the homeomorphism group of the Cantor set C. It is generated by involutions [2, Section 12] and, if the metric on C is ignored, V can be viewed somewhat as a “Coxeter group” of permutations of C. In [4] we find presentations for BV and V that differ only in that the presentation for V has relations of the form x = 1 that are not present in the presentation for BV . Thus BV can be thought of as an “Artinification” of V . Our motivation for creating BV is a relationship between BV and the Thompson’s groups F and V on the one hand, and categories with multiplication on the other. Given a category C with multiplication, an isomorphism expressing associativity up to equivalence, and perhaps an isomorphism expressing commutativity up to equivalence, there are groups and epimorphisms

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/405e92ef17993504a383d4941221a3946b4afa42

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Citations in the corpus

The braided Thompson's groups are of type F∞ (2012, methodology): “ one is obtained from the other by a sequence of reductions or expansions. The multiplication operation on the equivalence classes is dened the same way as for V. It is a well dened group operation [Bri07]. 6 K.-U. BUX, M. G. FLUCH, M. MARSCHLER, S. WITZEL, AND M. C. B. ZAREMSKY Denition 1.3. The br”
The braided Ptolemy-Thompson group is finitely presented (2005, background): “raids B∞ on inﬁnitely many strands. Notice that an extension BV of the larger Thompson group V by a pure braid group on inﬁnitely many strands has been recently considered by M. Brin and P. Dehornoy ([4, 5, 12, 13]). It constitutes the planar counterpart of the group B, in which the Cantor surface i”