The current semester I'm teaching MTH 222T/331T (MTWRF 4:10pm - 5:00 pm), and MTH 491/591 MTH 491/591 (TR 2:15 pm - 3:30 pm)
RESEARCH INTERESTS:
My research focuses on the interplay
between geometry, topology, and algebraic K-theory. Here are all my completed projects. All my work presented here is partly supported by the National Science
Foundation under the grant DMS-0805605 (2008-20011).
(Last Update: August 10, 2009)
Submitted papers:
- Algebraic K-theory of hyperbolic reflection groups,
(joint with B. Magurn and J.-F. Lafont). 35 pages as a
preprint,
pdf. A 3-dimensional hyperbolic reflection group is a Coxeter group arising as a
lattice in the isometry
group of hyperbolic 3-space, with fundamental domain a geodesic finite volume geodesic polyhedron P.
Building on
our previous work (the case where P was a tetrahedron), we provide formulas for the
loweralgebraic K-theory of the integral group ring of all the 3-dimensional hyperbolic reflection groups.
The expressions for the lower algebraic K-theory end up depending solely on the internal dihedral angles
between the faces of the polyhedron P. In particular, the computation of the lower algebraic K-theory of
such groups reduces to the computation of the lower algebraic K-theory of dihedral groups, as well as products of dihedral groups with the cyclic group of order two.
Published papers:
- Splitting formulas for certain Waldhausen Nil-groups,
(joint with J.-F. Lafont).
pdf 16 pages as a preprint. Final version in J. London Math. Soc. 79 (2009), pgs. 309-322.
For a group G that splits as an amalgamation of A and B over a common subgroup C, there is an
associated Waldhausen Nil-group, measuring the "failure" of Mayer-Vietoris for algebraic K-theory. Assume that (1) the amalgamation is acylindrical, and (2) the groups A,B,G satisfy the Farrell-Jones isomorphism conjecture. Then we show that the Waldhausen Nil-group splits as a direct sum of Nil-groups associated to certain (explicitly describable) infinite virtually cyclic subgroups of G. We
note that a special case of an acylindrical amalgamation includes any amalgamation over a finite
group. Taken in combination with recent work by several mathematicians (J. Davis, Q. Khan, A. Ranicki, H. Reich, and F. Quinn), this completely reduces (modulo the FJ-isomorphism conjecture) the computation of Waldhausen Nil-groups associated to acylindrical amalgamations to the considerably easier computation of Farrell Nil-groups associated with various virtually cyclic subgroups.
- Algebraic K-theory of hyperbolic 3-simplex reflection groups,
(joint with Jean-Francois Lafont), pdf.
33 pages as a preprin. Final version in Comment. Math. Helv. 84 (2009), pgs. 297-337.
A hyperbolic 3-simplex reflection group is a Coxeter group arising as a lattice in the isometry
group of hyperbolic 3-space, with fundamental domain a geodesic simplex (possibly with some
ideal vertices). The classification of these groups is known, and there are exactly 9 cocompact
examples, and 23 non-cocompact examples. We provide a complete computation of the lower
algebraic K-theory of the integral group ring of all the hyperbolic 3-simplex reflection groups. In an Addendum to our paper,
C. Weibel
provided a refinement of some of our computations, by
explicitly computing some of the Nil groups that appear in our expressions.
- Relating the Farrell Nil-groups to the Waldhausen Nil-groups,
(joint with Jean-Francois Lafont),
pdf. 10 pages as a preprint. Final version in Forum Math. 20 (2008), pgs. 445-455.
We prove that the Waldhausen
Nil-groups associated to a virtually cyclic group that surjects onto
the infinite dihedral group vanishes if and only if
the Farrell Nil-group associated to the canonical index two subgroup is trivial.
The proof uses the transfer map to establish one direction, and uses controlled pseudo-isotopy
techniques of Farrell-Jones to establish the reverse implication.
- Relative hyperbolicity, classifying spaces, and lower algebraic K-theory,
(joint with Jean-Francois Lafont),
pdf. 28 pages as a preprint. Ffinal version in
Topology 46 (2007), pgs. 527-553.
For G a relatively hyperbolic group, we provide a recipe for constructing a model for the
(2007), pgs. 527-553".
For G a relatively hyperbolic group, we provide a recipe for constructing a model for the
universal space among G-spaces with isotropy in the family of virtually cyclic subgroups of G.
For G a Coxeter group acting as a non-uniform lattice on hyperbolic 3-space, we construct
the classifying space explicitly, resulting in an 8-dimensional classifying space. We use
the classifying space we obtain to compute the lower algebraic K-theory for one of these
Coxeter groups.
- Erratum to The lower algebraic K-theory of Gamma_3,
pdf. 2 pages as a preprint. Ffinal version in K-theory
38 (2007), pgs. 85-86.
The lower algebraic K-theory of Gamma_3 presented in the Main
Theorem of Or04 is incomplete. In this erratum we present the correct
version of this theorem.
- The Lower Algebraic K-theory of Gamma_3, pdf. 20 pages
as a preprint. Final version in K-theory 32 no.4 (2004), pgs.
331-355).
We explicitly compute the lower algebraic K-theory of Gamma_3 a discrete subgroup of the group
of isometries of hyperbolic 3-space.
Work in progress:
The following projects are in various stages of typing. Preprints will be available
as soon as they get completed. The descriptions below reflect, to the best of my knowledge,
the results that will be appearing in the completed papers. The projects are organized
roughly according to proximity to completion (closest to finished are at the top of the list)
- Three-dimensional crystallographic groups and algebraic K-theory In a joint
work with Dan Farley, we are working on
computing the lower algebraic K-theory of three-dimensional crystallographic groups.
Alves and Ontaneda, give a simple formula for
for the Whitehead groupcrystallographic group G in terms of the Whitehead groups of the virtually
infinite cyclic subgroups of G. The main goal in this project is to obtain explicit computations for
K_0(ZG) and K_{-1}(ZG) for these groups,
as Pearson
did in the 2-dimensional case.
- Kleinian groups: lattice retracts, accessibility, and the Farrell-Jones isomorphism
conjectures,
(joint with D. Juan-Pineda, Jean-Francois Lafont, and D. Vavrichek).
Using some of the spectacular recent work in 3-manifold theory, we show that various isomorphism conjectures known to hold for lattices in the isometry group of hyperbolic 3-space actually hold for the broader class of Kleinian groups. In previous work, I'd developed techniques with Lafont for computing the lower algebraic K-theory of lattices inside the isometry group of hyperbolic 3-space; we also show that these techniques can now be extended to the setting of Kleinian groups.
[This paper still needs some work. We can currently deal with the case of Kleinian groups that are 1-ended and do not split over any 2-ended subgroup. We are working on removing the "does not split over 2-ended subgroup" hypothesis.]
- The lower algebraic K-theory of Gamma_4. Let Gamma_4 be the group of integral, positive, Lorentzian
5 x 5 matrices; this group is a non-cocompact, 4-simplex, hyperbolic reflection group. In this project we are currently working on getting an explicitly computation of the lower algebraic K-theory of the integral group ring of Gamma_4.
What is Topology, anyway?
Here is
Robert Bruner's answer.
My beautiful Kids
Oscar & Tomas |
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