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 636
Phone: (614)-292-1849

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.

Here's a link to information on my upcoming travel plans, visitors, and other events.

And here's a pretty hilarious music video (which I got from my youngest brother... hmmm... wonder why he forwarded it to me?)

Former Students:

Current Students:

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 grants DMS-0606002 (2006-2009), DMS-0906483 (2009-2012), DMS-1207782 (2012-2015), and by an Alfred P. Sloan Research Fellowship (2008-2012).

You can view/hide the individual papers abstracts by clicking on the appropriate link. Alternatively you can Show all Abstracts | Hide all Abstracts

If you find yourself enjoying this sort of math, you might be interested in having a look at the work of some of my collaborators: G. Arzhantseva, D. Constantine, M. Davis, F.T. Farrell, J. Fowler, S. Francaviglia, R. Frigerio, T. Januszkiewicz, D. Juan-Pineda, B. Magurn, S. Millan-Vossler, A. Minasyan, I. Ortiz, S. Pallekonda, Ch. Pittet, E. Prassidis, R. Roy, R. Sánchez-García, B. Schmidt, and A. Sisto.

Submitted papers:

• Aspherical manifolds that cannot be triangulated (joint with M. Davis and J. Fowler)

  Abstract. 9 pages as a preprint (April 2013)

  We apply hyperbolization techniques to construct, in each dimension >5, closed aspherical manifolds that cannot be triangulated. This complements a construction of 4-dimensional examples by Davis-Januszkiewicz. The question remains open in dimension =5.

• Marked length rigidity for one dimensional spaces (joint with D. Constantine)

  Abstract. 39 pages as a preprint (September 2012)

  We prove that for compact, non-contractible, one dimensional geodesic spaces, a version of the marked length spectrum conjecture holds. For a compact one dimensional geodesic space X, we define a subspace Conv(X). When X is non-contractible, we show that X deformation retracts to Conv(X). If two such spaces X, Y have the same marked length spectrum, we prove that Conv(X) and Conv(Y) are isometric to each other.

• Rigidity of high dimensional graph manifolds (joint with R. Frigerio and A. Sisto)

  Abstract. 147 + xv pages as a preprint (July 2011).

  We define the class of high dimensional graph manifolds. These are compact smooth manifolds supporting a decomposition into finitely many pieces, each of which is diffeomorphic to the product of a torus with a finite volume hyperbolic manifold with toric cusps. The various pieces are attached together via affine maps of the boundary tori. We require all the hyperbolic factors in the pieces to have dimension at least 3. Our main goal is to study this class of graph manifolds from the viewpoint of rigidity theory. We show that, in high dimensions, the Borel conjecture holds for our graph manifolds. We also show that smooth rigidity holds within the class: two graph manifolds are homotopy equivalent if and only if they are diffeomorphic. We introduce the notion of irreducible graph manifolds, which form a subclass which has better coarse geometric properties. We establish some structure theory for finitely generated groups which are quasi-isometric to the fundamental group of an irreducible graph manifold: any such group has a graph of groups splitting with strong constraints on the edge and vertex groups. Finally, we prove that in every dimension >3 there exist examples of irreducible graph manifolds which do not support any locally CAT(0) metric.

Accepted papers:

• Isomorphism versus commensurability for a class of finitely presented groups (joint with G. Arzhantseva and A. Minasyan)

  Abstract. 13 pages as a preprint (September 2011), to appear in J. Group Theory.

  We construct a class of finitely presented groups where the isomorphism problem is solvable but the commensurability problem is unsolvable. Conversely, we construct a class of finitely presented groups within which the commensurability problem is solvable but the isomorphism problem is unsolvable. These are the first examples of such a contrastive complexity behavior with respect to the isomorphism problem.

Published papers:

2012

• Rational equivariant K-homology of low dimensional groups (joint with I. Ortiz and R. Sánchez-García)

  Abstract. 33 pages as a preprint. Final version in Clay Math. Proceedings 16 (2012), pgs. 131-164. The volume is entitled Topics in Noncommutative Geometry, Proceedings of the 3rd Winter School at the Luis Santaló-CIMPA Research School (Buenos Aires, 2010).

  We consider groups G which have a cocompact 3-manifold model for the classifying space for proper G-actions. We provide an algorithm for computing the rationalized equivariant K-homology of the classifying space. Under the additional hypothesis that the G-action on the 3-dimensional model is smooth, the Baum-Connes conjecture holds, and the rationalized K-homology groups of the classifying space coincide with the rationalized topological K-theory of the reduced C*-algebra of G. We illustrate our algorithm on several concrete examples.

• Comparing semi-norms on homology (joint with Ch. Pittet)

  Abstract. 13 pages as a preprint. Final version in Pacific J. Math. 259 (2012), pgs. 373-385.

  We compare the l¹-seminorm and the manifold seminorm on integral homology classes. The l¹-seminorm is always bounded above by the manifold seminorm. We explain how it easily follows from work of Crowley & Löh that in degrees distinct from three, these two seminorms in fact coincide. We compute the simplicial volume of the 3-dimensional Tomei manifold and apply Gaĭfullin's desingularization to establish the existence of a constant (approximately equal to 0.0115416), with the property that for any degree three homology class, the l¹-seminorm is bounded below by the constant times the manifold seminorm.

• 4-dimensional locally CAT(0)-manifolds with no Riemannian smoothings (joint with M. Davis and T. Januszkiewicz)

  Abstract. 20 pages as a preprint. Final version in Duke Math. Journal 161 (2012), pgs. 1-28.

  We construct examples of 4-dimensional manifolds M supporting a locally CAT(0)-metric, whose universal cover X satisfy Hruska's isolated flats condition, and contain 2-dimensional flats F with the property that the boundary at infinity of F defines a nontrivial knot in the boundary at infinity of X. As a consequence, we obtain that the fundamental group of M cannot be isomorphic to the fundamental group of any closed Riemannian manifold of nonpositive sectional curvature. In particular, M is a locally CAT(0)-manifold which does not support any Riemannian metric of nonpositive sectional curvature.

2011

• Algebraic K-theory of virtually free groups (joint with D. Juan-Pineda, S. Millan-Vossler, and S. Pallekonda)

  Abstract. 22 pages as a preprint. Final version in Proc. Roy. Soc. Edinburgh Sect. A 141 (2011), pgs. 1295-1316.

  We provide a general procedure for computing the algebraic K-theory of finitely generated virtually free groups. The procedure describes these groups in terms of (1) the algebraic K-theory of various finite subgroups, and (2) various Farrell Nil-groups. We illustrate the process by carrying out the computation for several interesting classes of examples. The first two classes serve as a check on the method, and show that our algorithm recovers results already existing in the literature. The last two classes of examples yield new computations.

2010

• Large scale detection of half-flats in CAT(0)-spaces (joint with S. Francaviglia)

  Abstract. 21 pages as a preprint. Final version in Indiana Univ. Math. J. 59 (2010), pgs. 395-415.

  For a k-flat F inside a locally compact CAT(0)-space X, we identify various conditions that ensure that F bounds a (k+1)-dimensional half flat in X. Our conditions are formulated in terms of the ultralimit of X. As applications, we obtain (1) constraints on the behavior of quasi-isometries between tocally compact CAT(0)-spaces, (2) constraints on the possible non-positively curved Riemannian metrics supported by certain manifolds, and (3) a correspondence between metric splittings of a complete, simply connected, non-positively curved Riemannian manifold and the metric splittings of its asymptotic cones. Furthermore, combining our results with the Ballmann, Burns-Spatzier rigidity theorem and the classical Mostow rigidity theorem, we also obtain (4) a new proof of Gromov's rigidity theorem for higher rank locally symmetric spaces.

• Lower algebraic K-theory of certain reflection groups (joint with B. Magurn and I. Ortiz).

  Abstract. 35 pages as a preprint. Final version in Math. Proc. Cambridge Philos. Soc. 148 (2010), pgs. 193-226.

  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, in terms of the combinatorics of the polyhedron P. As part of the computation, we provide number theoretic formulas for some of the lower algebraic K-groups of dihedral groups, as well as products of dihedral groups with the cyclic group of order two.

2009

• A boundary version of Cartan-Hadamard and applications to rigidity

  Abstract. 30 pages as a preprint. Final version in J. Topol. Anal. 1 (2009), pgs. 431-459.

  We show that in dimensions distinct from five, any two compact, negatively curved Riemannian manifolds with non-empty, totally geodesic boundary, have universal covers with homeomorphic boundaries at infinity. We show that in any given dimension, the diffeomorphism type of the universal cover of a compact, non-positively curved Riemannian manifolds with totally geodesic boundary is completely determined by the number of boundary components of the universal cover. We also show that the number of boundary components is either 0, 2, or 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).

• A note on strong Jordan separation

  Abstract. 8 pages as a preprint. Final version in Publ. Mat. 53 (2009), pgs. 515-525.

  We provide a strengthening of Jordan separation, to the setting of maps from a compact topological space X into a sphere, where the source space X is not necessarily a codimension one sphere, and the map is not necessarily injective.

• Splitting formulas for certain Waldhausen Nil-groups (joint with I. Ortiz)

  Abstract. 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.

• Lower algebraic K-theory of hyperbolic 3-simplex reflection groups (joint with I. Ortiz)

  Abstract. 33 pages as a preprint. 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.

• Involutions of negatively curved groups with wild boundary behavior (joint with F.T. Farrell)

  Abstract. 19 pages as a preprint. Final version in the Hirzebruch special issue of Pure Appl. Math. Q. 5 (2009), pgs. 619-640.

  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.

2008

• Relating the Farrell Nil-groups to the Waldhausen Nil-groups (joint with I. Ortiz)

  Abstract. 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. Here is a minor correction to the paper.

• Construction of classifying spaces with isotropy in prescribed families of subgroups

  Abstract. 4 pages as a preprint. Final version in L'Enseign. Math. (2) 54 (2008), pgs. 131-134.

  This is a short collection of open problems, to honor the retirement of Guido Mislin.

2007

• Relative hyperbolicity, classifying spaces, and lower algebraic K-theory (joint with I. Ortiz)

  Abstract. 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

  Abstract. 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)

  Abstract. 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

  Abstract. 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)

  Abstract. 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)

  Abstract. 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)

  Abstract. 15 pages as a preprint. Final version in Acta Math. 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

  Abstract. 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 not necessarily injective maps from an (n-1)-sphere to an n-sphere. 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)

  Abstract. 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)

  Abstract. 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 (**). 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 (**). Our second theorem shows that condition (**) 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

  Abstract. 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)

  Abstract. 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.

Papers being revised:

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).

Papers not intended for publication:

• Asymptotic cones, bi-Lipschitz ultraflats, and the geometric rank of geodesics (joint with S. Francaviglia)

  Abstract. 35 pages as a preprint, pdf.

  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.

  [A considerably stronger version of these results (using metric as opposed to Riemannian arguments) appears in my paper with Stefano: ``Large scale detection of half-flats in CAT(0)-spaces''. ]

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).

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

• Length spectra and (almost) arithmetic progressions (joint with D.B. McReynolds). Summary.
  In analogy to the Green-Tao theorem, we search for arithmetic progressions in the primitive length spectrum of negatively curved spaces. We show that, within the space of negatively curved metrics on a fixed smooth manifold M, there is a dense G-delta set of metrics whose primitive length spectrum contains no arithmetic progression of length greater than 2. We introduce the notion of an almost arithmetic progression, and show that in any compact locally CAT(-1) space, the corresponding primitive length spectrum contains almost arithmetic progressions of arbitrarily long length. Finally, we consider the case where genuine arithmetic progressions exist. For real hyperbolic manifolds whose fundamental groups are arithmetic lattices, we show that every number in the primitive length spectrum lies in arbitrarily long arithmetic progressions. We conjecture this property characterizes arithmeticity of the lattice.

  [This paper still needs some work. We have outlines for all the results mentioned here, but the details need to be carefully checked.]

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