Jean-François Lafont

Department of Mathematics
The Ohio State University
100 Math Tower
231 West 18th Avenue
Columbus, OH 43210-1174, USA

E-mail: jlafont@math.ohio-state.edu
Office: MW 510
Phone: (614)-292-5814

Other Information:

I finished my doctorate at the University of Michigan in 2002, under the guidance of Ralf Spatzier. I was a postdoc at S.U.N.Y. Binghamton from 2002-2005, where I went to collaborate with Tom Farrell.

This quarter, I am serving as organizer of the Topology Seminar.

Information on my upcoming travel plans can be found here.

During the next year, I will be teaching the Math 866-867-868 sequence, which is a year long course in

Differential Topology

RESEARCH INTERESTS:

My research focuses on the interplay between geometry, topology, and group theory, particularly in the presence of non-positive curvature. Here are all of my completed projects (in reverse chronological order). The work done here is partly supported by the National Science Foundation under grant DMS-0606002 (2006-2009), and by an Alfred P. Sloan Research Fellowship (2008-2010).

Submitted papers:

(Last Update: April 3rd, 2008)

• Splitting formulas for certain Waldhausen Nil-groups, (joint with I. Ortiz), pdf. 12 pages as a preprint. 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.

Papers being revised:

(Last Update: April 3rd, 2008)

These papers are complete, but are currently being revised in order to improve the results they contain. Be aware that the results in these papers, while correct, are definitely not optimal. I also include [in brackets] the improvements I believe can be done on the existing results (and which are currently being worked on).

• Asymptotic cones, bi-Lipschitz ultraflats, and the geometric rank of geodesics, (joint with S. Francaviglia), pdf. 35 pages as a preprint. Given a geodesic inside a simply-connected, complete, non-positively curved Riemannian (NPCR) manifold M, we get an associated geodesic inside the asymptotic cone Cone(M). Under mild hypotheses, we show that if the latter is contained inside a bi-Lipschitz flat, then the original geodesic supports a non-trivial, orthogonal, parallel Jacobi field. As applications we obtain (1) constraints on the behavior of quasi-isometries between complete, simply connected, NPCR manifolds, and (2) constraints on the NPCR metrics supported by certain manifolds, and (3) a correspondence between metric splittings of complete, simply connected NPCR manifolds, and metric splittings of its asymptotic cones. Furthermore, combining our results with the Ballmann, Burns-Spatzier rank rigidity theorem and the classic Mostow rigidity, we also obtain (4) a new proof of Gromov's rigidity theorem for higher rank locally symmetric spaces. [We're trying to improve the main theorem to allow for determining the exact rank of geodesics under certain hypotheses on the ultralimit.]

• A boundary version of Cartan-Hadamard and applications to rigidity, pdf. 15 pages as a preprint. We show that in dimensions distinct from five, any two compact, negatively curved Riemannian manifolds with non-empty, totally geodesic boundary, have (1) universal covers that are homeomorphic, and (2) fundamental groups with homeomorphic boundaries at infinity. As an application, we show that simple, thick, negatively curved P-manifolds of dimension greater than five are topologically rigid. We discuss various corollaries of topological rigidity (diagram rigidity, weak co-Hopf property, Nielson realization problem). [I have already removed the restriction on the dimension, and improved the curvature constraint from "negative" to "non-positive" (when zero curvature is present, there are some additional possibilities). I'm currently working on improving the conclusion in (1) from "homeomorphic" to "diffeomorphic", which I can easily do in high dimension by smoothing theory. I'm trying to give a simpler proof that also works in low dimensions.]

Accepted papers:

• Lower algebraic K-theory of hyperbolic 3-simplex reflection groups, (joint with I. Ortiz), pdf. 33 pages as a preprint, to appear in Comment. Math. Helv. 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 I. Ortiz), pdf. 10 pages as a preprint, to appear in Forum Math. 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.

• Involutions of negatively curved groups with wild boundary behavior, (joint with F.T. Farrell), pdf. 19 pages as a preprint, to appear in a special issue of Pure and Applied Math Quarterly in honor of Prof. F. Hirzebruch. 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 an algebraic criterion for knottedness of tame codimension two spheres in high dimensional (>5) spheres.

Published papers:

2007

• Relative hyperbolicity, classifying spaces, and lower algebraic K-theory, (joint with I. Ortiz), pdf. 28 pages as a preprint (final version in Topology 46 (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.

• Rigidity of hyperbolic P-manifolds: a survey, pdf. 11 pages as a preprint (final version in Geom. Dedicata 124 (2007), pgs. 143-152). In this survey paper, we outline the proofs of the rigidity theorems for simple, thick, hyperbolic P-manifolds found in three of our earlier papers ("Diagram rigidity", "Strong Jordan separation" and "Rigidity results").

• Blocking light in compact Riemannian manifolds, (joint with B. Schmidt), pdf. 19 pages as a preprint (final version in Geom. Topol. 11 (2007), pgs. 867-887). We study closed Riemannian manifolds (M,g) for which the light from any given point can be shaded away from any other point by finitely many point shades in M. Closed flat Riemannian manifolds are known to have this finite blocking property. We conjecture that all such metrics are flat. Using entropy considerations, we verify this conjecture amongst metrics with non-positive sectional curvatures. Using the same approach, K. Burns and E. Gutkin have independently obtained this result. Additionally, we show that compact quotients of Euclidean buildings have the finite blocking property. In a different direction, we conjecture that closed Riemannian manifolds with the same blocking properties as compact rank one symmetric spaces are necessarily isometric to a compact rank one symmetric space. We include some results providing evidence for this conjecture.

• Diagram rigidity for geometric amalgamations of free groups, pdf. 16 pages as a preprint (final version in J. Pure Appl. Algebra 209 (2007), pgs. 771-780). 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.

• A note on characteristic numbers of non-positively curved manifolds, (joint with R. Roy), pdf. 11 pages as a preprint (final version in Expo. Math. 25 (2007), pgs. 21-35). In this expository paper, we provide vanishing/non-vanishing results for characteristic numbers of non-positively curved Riemannian manifolds. In the locally symmetric case we give a very simple proof of the Hirzebruch proportionality principle for Pontrjagin numbers. We also exhibit vanishing of some characteristic numbers for the Gromov-Thurston examples of negatively curved manifolds. A byproduct of our argument is a simple constructive proof of Rohlin's Theorem: that every compact orientable 3-manifold bounds orientably. Various topological consequences are discussed, and some new applications are given.

2006

• On submanifolds in locally symmetric spaces of non-compact type, (joint with B. Schmidt), pdf. 16 pages as a preprint (final version in Algebr. Geom. Topol. 6 (2006), pgs. 2455-2472). 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 Y is homologically non-trivial in X. In low dimensions (relative to the dimension of X), our sufficient condition is also necessary. We provide conditions under which there exist a tangential map of pairs from a finite cover of the pair (X,Y) to the non-negatively curved dual pair of spaces.

• Simplicial volume of closed locally symmetric spaces of non-compact type, (joint with B. Schmidt), pdf. 15 pages as a preprint (final version in Acta Mathematica 197 (2006), pgs. 129-143). 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.

• Strong Jordan separation and applications to rigidity, pdf. 26 pages as a preprint (final version in J. London Math. Soc. 73 (2006), pgs. 681-700). We establish Mostow type rigidity and quasi-isometric 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". Strong Jordan separation was recently used by T. Iwaniec and J. Onninen in their work on quasiconformal hyperelasticity.

• Roundness properties of groups, (joint with E. Prassidis), pdf. 22 pages as a preprint (final version in Geom. Dedicata 117 (2006), pgs. 137-160). 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.

2005

• EZ-structures and topological applications, (joint with F.T. Farrell), pdf. 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.

2004

• Rigidity results for certain 3-dimensional singular spaces and their fundamental groups, pdf. 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.

• Finite automorphisms of negatively curved Poincare Duality groups, (joint with F.T. Farrell), pdf. 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.

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. Where possible, I state [in brackets] the work that still remains to be done on the various projects. The projects are organized roughly according to proximity to completion (closest to finished are at the top of the list).

• Algebraic K-theory of hyperbolic reflection groups, (joint with B. Magurn and I. Ortiz). 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 finite volume geodesic polyhedron P. Building on our previous work (the case where P was a tetrahedron), we provide formulas for the lower algebraic 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. This allows us to write out explicit formulas for the Whitehead group. [This paper is almost done. We are still working on obtaining explicit formulas for the remaining lower K-groups.]

• Kleinian groups: lattice retracts, accessibility, and the Farrell-Jones isomorphism conjectures, (joint with D. Juan-Pineda, I. Ortiz, 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 Ortiz 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.]

• Simplicial volume of finite volume locally symmetric spaces of non-compact type. (joint with M. Bucher-Karlsson) We show that certain finite volume locally symmetric spaces of non-compact type have positive simplicial volume. Some new applications we obtain are: (1) these manifolds do not collapse, (2) the minimal volume of these manifolds are positive, and (3) a degree theorem for proper maps into these manifolds. [We have a proof in the R-rank 1 case, and are currently working on extending the result to the case of Q-rank 1 lattices in higher rank symmetric spaces. In contrast, R. Sauer and C. Löh have an argument showing that lattices of Q-rank > 1 have zero simplicial volume. This project has been in stasis for the last year.]

• Rigidity of manifolds with(out) nonpositive curvature. . We define a new class of closed manifolds which are built up from non-positively curved pieces. We show that this class of manifolds contains some examples which do not support any locally CAT(0) metric. With some additional hypotheses, we show that these manifolds exhibit various types of rigidity (quasi-isometric, topological, smooth). Finally, we give a complete (algebraic) criterion for whether two such manifolds have isomorphic fundamental groups. [This paper is still in a very preliminary form; I'm still working on establishing the rigidity results. I have outlines for all the results stated above, as well as detailed proofs for both the examples with no locally CAT(0) metrics and the solution to the isomorphism problem.]

• 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 once I get around to working on this again.]

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