We study the group of orientation-preserving homeomorphisms of the real line R which are piecewise-linear with respect to a finite subdivision of R; we denote this group PLF(R). Our main results are presentations of PLF(~) and certain of its subgroups; a proof that PLF(R) contains no free subgroups of rank greater than 1; and a proof that PLF(R) satisfies no laws. In the process, we construct a sequence of finitely-presented subgroups G(p) of PLF(R) which contain no free subgroups of rank greater than 1 and satisfy no laws. We also briefly consider the full group of piecewise-linear homeomorphisms of R and the group of piecewise-linear homeomorphisms of the circle S 1.
We construct a "higher dimensional" version 2V of Thompson's group V. Like V
it is an infinite, finitely presented, simple subgroup of the homeomorphism
group of the Cantor set, but we show that it is not isomorphic to V by showing
that the actions on the Cantor set are not topologically conjugate: 2V has an
element with "chaotic" action, while V cannot have such an element. A theorem
of Rubin is then applied which shows that for these two groups, isomorphism
would imply topological conjugacy.Comment: 27 pages To appear in Geometriae Dedicat
The pictures in (2) simply express the fact that certain splitting and recombining operations are inverses of each other.We are in a position to multiply pictorally. As in the braid group, the product uv of u and v is drawn by putting u over v. If u is the element
The Zappa-Szép product of a pair of groups generalizes the semidirect product in that neither factor is assumed to be normal in the result. We extend the applicability of the Zappa-Szép product to multiplicative structures more general than groups with emphasis on categories and monoids. We also explore the preservation of various properties of the multiplication under the Zappa-Szép product.
BRINProof. It suffices to consider a right identity a for A. There is a left identity b for a that is a left identity for A. Now ba equals both a and b, so a is both a left and a right identity for A. Now that a = b, we have aDa and a 2 = a.
In a previous paper, we defined a higher dimensional analog of Thompson's
group V, and proved that it is simple, infinite, finitely generated, and not
isomorphic to any of the known Thompson groups. There are other Thompson groups
that are infinite, simple and finitely presented. Here we show that the new
group is also finitely presented by calculating an explicit finite
presentation.Comment: 35 pages, to appear in J. Algebr
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