The algebra of strand splitting. I. A braided version of Thompson’s group V
نویسنده
چکیده
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
منابع مشابه
The Algebra of Strand Splitting Ii: a Presentation for the Braid Group on One Strand
In [3], we give descriptions of a braided version BV of Thompson’s group V as well as a group B̂V that contains BV as a subgroup and that is somewhat easier to work with. The paper [3] contains both geometric and algebraic descriptions of these two groups and shows that for each group the two descriptions are of isomorphic groups. An infinite presentation for B̂V is also given in [3]. The current...
متن کاملPure Braid Subgroups of Braided Thompson’s Groups
We describe pure braided versions of Thompson’s group F . These groups, BF and B̂F , are subgroups of the braided versions of Thompson’s group V , introduced by Brin and Dehornoy. Unlike V , elements of F are order-preserving self-maps of the interval and we use pure braids together with elements of F thus preserving order. We define these groups and give normal forms for elements and describe i...
متن کاملMetric Properties of the Braided Thompson’s Groups
Braided Thompson’s groups are finitely presented groups introduced by Brin and Dehornoy which contain the ordinary braid groups Bn, the finitary braid group B∞ and Thompson’s group F as subgroups. We describe some of the metric properties of braided Thompson’s groups and give upper and lower bounds for word length in terms of the number of strands and the number of crossings in the diagrams use...
متن کاملFiniteness Properties of Some Groups of Local Similarities
Hughes has defined a class of groups, which we call FSS (finite similarity structure) groups. Each FSS group acts on a compact ultrametric space by local similarities. The best-known example is Thompson’s group V . Guided by previous work on Thompson’s group V , we establish a number of new results about FSS groups. Our main result is that a class of FSS groups are of type F∞. This generalizes ...
متن کاملBi-orderings on Pure Braided Thompson’s Groups
In this paper it is proved that the pure braided Thompson’s group BF admits a bi-order, analog to the bi-order of the pure braid groups.
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تاریخ انتشار 2004