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Jean-Francois Lafont
Department of Mathematical Sciences
Binghamtom University
Binghamton, NY 13902-6000 USA

RESEARCH INTERESTS:

My research focuses on the interplay between geometry, topology, and group theory. In particular, understanding the relationships between properties of a group and the geometry of a "nice" space on which the group acts by isometries. Here are all of my completed projects (in reverse chronological order):

• Diagram rigidity for geometric amalgamations of free groups, dvi, ps. 16 pages as a preprint (submitted). We prove a topological rigidity result for simple, thick, hyperbolic P-manifolds of dimension 2: isomorphism of the fundamental groups implies homeomorphism of the P-manifolds. An immediate application is a diagram rigidity theorem for certain amalgamations of free groups: the direct limits of two such diagrams are isomorphic if and only if there is an isomorphism between the respective diagrams.

• Simplicial volume of closed locally symmetric spaces of non-compact type, (joint with B. Schmidt), dvi, ps. 15 pages as a preprint (submitted). We show that compact, locally symmetric spaces of non-compact type have positive simplicial volume. This gives a positive answer to a question that was first raised by Gromov in 1982. We provide a summary of results that are known to follow from positivity of the simplicial volume.

• Roundness properties of groups, (joint with E. Prassidis), dvi, ps. 22 pages as a preprint (submitted). We study topological/geometric consequences of roundness and generalized roundness (metric invariants introduced by P. Enflo with substantial applications in functional analysis). We show that any compact Riemannian manifold with non-trivial fundamental group has roundness =1. We show that proper geodesic spaces with roundness =2 are contractible. For a finitely generated group G, we define the roundness spectrum R[G], a subset of the positive reals. We show that R[G] always contains 1, and if G is infinite then R[G] is contained in the interval [1,2]. We show that, if G is a free group, then R[G] contains 2. We show that for the free abelian group on >1 generators, R[G]={1}. We prove that if a group G has the property that 1 is not in R[G], then G is a torsion group with every element of order 2, 3, 5, or 7. We point out that if a group has a presentation whose Cayley graph has generalized roundness >0, then it satisfies the coarse Baum-Connes conjecture (and hence, the strong Novikov conjecture). We show that for Kazhdan groups, every Cayley graph has generalized roundness =0.

• Strong Jordan separation and applications to rigidity, dvi, ps. 26 pages as a preprint (submitted). We establish Mostow type rigidity and quasi-isometry rigidity for simple, thick, hyperbolic P-manifolds of dimension >3. The main technical tool is a "strong" form of Jordan separation, that applies to maps from S^{n-1} to S^n that are not necessarily injective. This paper extends and completes the results in our previous paper "Rigidity results for certain 3-dimensional singular spaces and their fundamental groups".

• A note on characteristic numbers of non-positively curved manifolds, (joint with R. Roy), dvi, ps. 11 pages as a preprint (submitted). In this expository paper, we provide vanishing/non-vanishing results for characteristic numbers of non-positively curved locally symmetric spaces. Our arguments differs from the classical Hirzebruch proportionality principle (which is the standard computational tool for finding these numbers), but does make use of the non-negatively curved duals. We also exhibit vanishing of some characteristic numbers for the Gromov-Thurston examples of negatively curved manifolds. Various topological consequences are discussed, and some new applications are given.

• Involutions of negatively curved groups with wild boundary behavior, (joint with F.T. Farrell), dvi, ps. 19 pages as a preprint (submitted). For a totally geodesic subspace Y of a compact locally CAT(-1) space X, one has an embedding of the boundary at infinity of the universal cover of Y into the boundary at infinity of the universal cover of X. In the case where the boundaries at infinity are spheres whose dimensions differ by two, we show that if the embedding is tame, it is unknotted. We give examples of pairs (X,Y) where the embedding is indeed knotted. In our examples, the embedded codimension two sphere is the fixed point set of a naturally defined involution of the ambient sphere. In passing, we also give a complete criterion for knottedness of tame codimension two spheres in high dimensional (>5) spheres.

• Rigidity results for certain 3-dimensional singular spaces and their fundamental groups, dvi, ps. 23 pages as a preprint (final version in Geom. Dedicata 109 (2004), pgs. 197-219). We introduce hyperbolic P-manifolds, which are certain non-positively curved metric spaces having a stratification by compact hyperbolic manifolds with totally geodesic boundary. For simple, thick, 3-dimensional hyperbolic P-manifolds, we give a topological criterion to recognize boundary points corresponding to lower dimensional strata. As a consequence of this main result, we obtain a version of Mostow rigidity for these spaces, as well as quasi-isometric rigidity for their fundamental groups.

• EZ-structures and topological applications, (joint with F.T. Farrell), dvi, ps. 19 pages as a preprint (final version in Comment. Math. Helv. 80 (2005), pgs. 103-121). We extend Bestvina's notion of a Z-structure to that of an EZ-structure, and extend Farrell-Hsiang's condition (*) to condition (*_\Delta). Examples of groups having an EZ-structure include delta hyperbolic groups and CAT(0) groups. Our first theorem shows that groups having an EZ-structure automatically satisfy condition (*_\Delta). Our second theorem shows that condition (*_\Delta) implies a version of the Novikov conjecture. Our third theorem restricts to the case of delta hyperbolic groups G, and provides a lower bound for the homotopy groups of the spaces obtained by applying the stable topological pseudo-isotopy functor to the classifying space of G.

• Finite automorphisms of negatively curved Poincare Duality groups, (joint with F.T. Farrell), dvi, ps. 11 pages as a preprint (final version in Geom. Funct. Anal. 14 (2004), pgs. 283-294). We show that, for a finite p-group acting on a negatively curved Poincare Duality group over Z, the fixed subgroup is a Poincare Duality group over Z/p. We provide examples to show that the fixed subgroup might not even be a duality group over Z.

• Rigidity results for singular spaces, dvi , ps . 99 pages. This is my doctoral thesis.

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.

• Marked length rigidity for one dimensional spaces. We prove that for compact one dimensional geodesic spaces, a version of the marked length spectrum conjecture holds. This conjecture states that the lengths of closed geodesics ''essentially'' determines the space in question. WARNING: a preliminary version of this paper contained an error in the proof of Lemma 2.2. Since this lemma was used repeatedly in the rest of the paper (and is incorrect as stated), the paper needs substantial rewriting. A corrected version will be posted in the (hopefully) not too distant future.

• Homologically essential submanifolds in locally symmetric spaces of non-compact type. (joint with B. Schmidt) Given a connected, totally geodesic submanifold Y inside a compact locally symmetric space of non-compact type X, we provide a condition that ensures that there exists a finite cover of X, with the property that every connected lift of Y in that cover is homologically non-trivial.

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