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In an attempt to understand "finite height (or
finite depth) measured laminations," I have used certain ordered
abelian semi-groups and ordered semi-rings. The finite height
laminations are characterized by having transverse measures with values
in an appropriate ordered semi-rings. Using the ordered semi-ring
as a parameter space, I have defined a space of finite height measured
laminations in a given surface, generalizing William Thurston's
projective measured lamination space for a surface. The new space
is probably related to hyperbolic geometry on the surface, just as in
the classical case. In the most recent paper on this subject, I
give a wide variety of orderd algebraic structures which can be used to
give transverse structures to laminations in more general
settings. Associated to the transverse structures, we have
actions on certain trees. The trees can either be thought of as
metric spaces with metrics having values in the ordered algebraic
structure, or they can be thought of as order trees with measures on
segments having values in the ordered algebraic structure.
The methods have further applications in measure theory and probability theory, though the simplicity of the applications makes me suspect that they are known in some form.
My motivations for this research were originally more related to
3-manifold topology. Namely, I hope to obtain a better
understanding of the measured lamination space of a 3-manifold by
understanding a larger space of finite height measured
laminations. Also, in the study of automorphisms of
3-manifolds, it appears that there exist invariant lamination-like
objects of mixed dimension for automorphisms of 3-manifolds, and these
should have finite height measured structures.
There are three related papers on this subject:
Finite height lamination spaces for
Applications of ordered abelian semi-groups and ordered semi-rings. Math arXiv
Automorphisms of 3-manifolds
The most interesting automorphisms (self-homeomorphisms) of 3-manifolds occur in boundary-reducible and reducible manifolds. The paper Automorphisms of 3-dimensional handlebodies develops a theory of automorphisms of handlebodies and compression bodies analogous to the Nielsen-Thurston theory of autmorphisms of surfaces. In the theory, the analogue of the pseudo-Anosov automorphism is the generic automorphism of a handlebody or compression body, and generic automorphisms turn out, as one would expect, to be the most interesting and mysterious. Automorphisms of irreducible 3-manifolds with non-empty boundary can then be understood by combining the new methods with older methods in 3-manifold topology, involving the Jaco-Shalen-Johannson characteristic submanifold and Bonahon's characteristic compression body. Much remains to be understood about generic automorphisms, which are currently also being studied by former Rutgers graduate student Leonardo Navarro de Carvalho.
In work with Carvalho, the ideas used for the classification of automorphisms of handlebodies and compression bodies are extended to classify, in a certain sense, the automorphisms of any compact 3-manifold (satisfying the Thurston Geometrization Conjecture). The current form of the paper is not satisfactory, and we plan to revise it extensively before publishing.
Automorphisms of 3-dimensional handlebodies and compression bodies, Topology 41 (2002) 363-410: Postscript file PDF file
A classification of automorphisms of
3-manifolds, with Leonardo Navarro de Carvalho. Math
Mapping class groups of compression bodies and 3-manifolds
In work related to the classification of automorphisms of 3-manifolds, I obtained results concerning the mapping class associated to a compression bodies. This is applied to obtain results concerning the mapping class group of a reducible 3-manifold M. The posted manuscript has been changed extensively, and the goals for the final version have become more and more ambitious. This paper is helpful for making a classification of automorphisms (as described above) more natural and appealing.
Mapping class groups of compression bodies and
Charalampos Charitos and I have a new normal surface
for incompressible (but not necessarily boundary incompressible)
in irrecucible (but not necessarily boundary irreducible) compact
with boundary. It may be possible to apply this theory to obtain
better invariant lamination-like objects for generic automorphisms of
and compression bodies. One of the motivations for the study of
finite height measured laminations (above) is that these invariant
lamination-like objects should be finite height measured in the
Essential disks and semi-essential surfaces in 3-manifolds, with Charalampos Charitos. Essential disks and semi-essential surfaces in 3-manifolds, Topology Appl. 159 (2012), no. 8, 2174-2186. Math arXiv
Contact structures and contaminations in 3-manifolds
In a project with Jacek Swiatkowski (Wroclaw University) I am studying contact structures using branched surface techniques. A natural generalization of the contact structure and the confoliation is the contamination, an object which can be carried by a branched surface. Contact structures can be represented by contaminations. The goals of this research are 1) to understand contact structures combinatorially; 2) to understand the relationships between foliations, confoliations, contact structures, laminations, and contaminations.
A contamination carrying criterion for branched surfaces, with J. Swiatkowski, Ann. Global Anal. Geom. 34 (2008), no. 2, 135-152. Math arXiv
Correction: "A contamination carrying criterion for branched surfaces," with J. Swiatkowski, Ann. Global Anal. Geom. 44 (2013), no. 2, 137-150.
Contact structures, sigma-confoliations, and contaminations in
3-manifolds, with J. Swiatkowski, Commun. Contemp. Math. 11
(2009), no. 2, 201-264.. Math
Some Other Papers