Presentations of higher . . .

In [2] we introduce groups nV and n̂V for integers n ≥ 1 and also ωV and ω̂V where ω represents the natural numbers. These are all subgroups of the homeomorphism group of the Cantor set. The group 1V is a group known as the Thompson group V which is infinite, simple and finitely presented (see [6]). In [2], we show that 2V is infinite, simple and finitely generated and that it is not isomorphic to V , and in fact not isomorphic to any member of an infinite collection of infinite, simple, finitely presented groups that are also known as Thompson groups. In this paper, we put the group 2V on the same status as V and the other infinite, simple, finitely presented Thompson groups by calculating a finite presentation for 2V . The group 2V is a subgroup of the (non-simple) group 2̂V and we also calculate a finite presentation for 2̂V . This is done partly out of necessity since the group 2̂V is easier to work with than 2V and a fairly full analysis of 2̂V (tantamount to calculating a presentation) must be done before an analysis of 2V can be done. The group 2̂V is a group of fractions of a particularly nice monoid Π that is also introduced in [2]. The monoid Π can be thought of as a countable, monoid approximation to the little cubes operads (in dimension 2) of [9] and [1]. The significance of this observation is unclear to the author. We gain the presentation and understanding of 2̂V by first calculating a presentation for Π. In [2], we give generating sets for Π, 2̂V and 2V . In that paper, we also list relations that are satisfied by the elements, but do not show that the relations suffice to give presentations. In this paper, we perform the calculations that show that we have enough relations for a presentation. The analysis of the presentation will proceed by using geometric representations of the elements that distinguish between different elements, and normal forms for words in the generators that are derived from the geometric representations. The work comes in showing that the relations suffice to reduce an arbitrary word to one of the normal forms. One step will make use of the analysis of 2̂V by mapping 2̂V into its subgroup 2V and using knowlege of 2̂V to perform part of the normalization.