43 research outputs found
Higher Dimensional Thompson Groups
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 Free Group of Rank 2 is a Limit of Thompson's Group F
We show that the free group of rank 2 is a limit of 2-markings of Thompson's
group F in the space of all 2-marked groups. More specifically, we find a
sequence of generating pairs for F so that as one goes out the sequence, the
length of the shortest relation satisfied by the generating pair goes to
infinity.Comment: 19 pages, 19 figure
Automorphisms of generalized Thompson groups
We look at the automorphisms of Thompson type groups of piecewise linear
homeomorphisms of the real line or circle that use slopes that are integral
powers of a fixed integer n with n>2. We show that large numbers of "exotic"
automorphisms appear---automorphisms that are represented as conjugation by
non-PL homeomorphisms of the real line or circle. This is in contrast to the
n=2 case where no such automorphisms appear.Comment: DVI and Post-Script files onl
Commentary on Robert Riley's article "A personal account of the discovery of hyperbolic structures on some knot complements"
We give some background and biographical commentary on the postumous article
that appears in this [journal issue | ArXiv] by Robert Riley on his part of the
early history of hyperbolic structures on some compact 3-manifolds. A complete
list of Riley's publications appears at the end of the article.Comment: 5 page
Coherence of Associativity in Categories with Multiplication
The usual coherence theorem of MacLane for categories with multiplication
assumes that a certain pentagonal diagram commutes in order to conclude that
associativity isomorphisms are well defined in a certain practical sense. The
practical aspects include creating associativity isomorphisms from a given one
by tensoring with the identity on either the right or the left. We show, by
reinspecting MacLane's original arguments, that if tensoring with the identity
is restricted to one side, then the well definedness of constructed
isomorphisms follows from naturality only, with no need of the commutativity of
the pentagonal diagram. This observation was discovered by noting the
resemblance of the usual coherence theorems with certain properties of a
finitely presented group known as Thompson's group F. This paper is to be taken
as an advertisement for this connection.Comment: 8 pages, to appear in Journal of Pure and Applied Algebr