Searcharxiv⌕ Search

arXiv subjects

Jean-François Lafont

Publications and source records attributed to Jean-François Lafont.

At least 19 recordsLinked to original sources
arXiv2026

Small Cancellation for Random Branched Covers of Groups

We construct a random model for an $n$-fold branched cover of a finite acceptable $2$-complex $X$. This includes presentation $2$-complexes for finitely presented groups satisfying some mild conditions. For any $λ>0$, we show that as $n$ goes to infinity, a random branched cover asymptotically almost surely is homotopy equivalent to a $2$-complex satisfying geometric small cancellation $C'(λ)$. As a consequence the fundamental group of a random branched cover is asymptotically almost surely Gromov hyperbolic and has small cohomological dimension.

math.GR↗
arXiv2025

An explicit description of the Kähler-Einstein metrics of Guenancia-Hamenstädt

Fine and Premoselli (FP) constructed the first examples of manifolds that do not admit a locally symmetric metric but do admit a negatively curved Einstein metric. The manifolds here are hyperbolic branched covers like those used by Gromov and Thurston, and the construction of their model Einstein metric is a variation of the hyperbolic metric written in polar coordinates. Very recently, Guenancia and Hamenstädt (GH) proved the existence of the first examples of manifolds that are not locally symmetric but admit a negatively curved Kähler-Einstein metric. The GH metrics are realized on complex hyperbolic branched covers constructed by Stover and Toledo. In this article we generalize the construction of FP to the complex hyperbolic setting and show that this yields a negatively curved Einstein metric that asymptotically approaches the metric of GH.

math.DG↗
arXiv2025

Relative cubulation of relative strict hyperbolization

We prove that many relatively hyperbolic groups obtained by relative strict hyperbolization admit a cocompact action on a CAT(0) cubical complex. Under suitable assumptions on the peripheral subgroups, these groups are residually finite and even virtually special. We include some applications to the theory of manifolds, such as the construction of new non-positively curved Riemannian manifolds with residually finite fundamental group, and the existence of non-triangulable aspherical manifolds with virtually special fundamental group.

math.GR↗
arXiv2024

Special cubulation of strict hyperbolization

We prove that the Gromov hyperbolic groups obtained by the strict hyperbolization procedure of Charney and Davis are virtually compact special, hence linear and residually finite. Our strategy consists in constructing an action of a hyperbolized group on a certain dual CAT(0) cubical complex. As a result, all the common applications of strict hyperbolization are shown to provide manifolds with virtually compact special fundamental group. In particular, we obtain examples of closed negatively curved Riemannian manifolds whose fundamental groups are linear and virtually algebraically fiber.

math.GR↗
arXiv2019

Strong symbolic dynamics for geodesic flow on CAT(-1) spaces and other metric Anosov flows

We prove that the geodesic flow on a locally CAT(-1) metric space which is compact, or more generally convex cocompact with non-elementary fundamental group, can be coded by a suspension flow over an irreducible shift of finite type with Hölder roof function. This is achieved by showing that the geodesic flow is a metric Anosov flow, and obtaining Hölder regularity of return times for a special class of geometrically constructed local cross-sections to the flow. We obtain a number of strong results on the dynamics of the flow with respect to equilibrium measures for Hölder potentials. In particular, we prove that the Bowen-Margulis measure is Bernoulli except for the exceptional case that all closed orbit periods are integer multiples of a common constant. We show that our techniques also extend to the geodesic flow associated to a projective Anosov representation, which verifies that the full power of symbolic dynamics is available in that setting.

math.DS↗
arXiv2019

Sets of degrees of maps between SU(2)-bundles over the 5-sphere

We compute the sets of degrees of maps between principal $SU(2)$-bundles over $S^5$, i.e. between any of the manifolds $SU(2)\times S^5$ and $SU(3)$. We show that the Steenrod squares provide the only obstruction to the existence of a mapping degree between these manifolds, and construct explicit maps realizing each integer that occurs as a mapping degree. Added Erratum. After this manuscript was accepted for publication by Transformation Groups, Xueqi Wang pointed out a mistake in our paper. At the end of this arXiv version we add an erratum, where we correct the statement and the proof of Theorem 1.1.

math.AT↗
arXiv2019

The weak specification property for geodesic flows on CAT(-1) spaces

We prove that the geodesic flow on a compact locally CAT(-1) space has the weak specification property, and give various applications. We show that every Hölder potential on the space of geodesics has a unique equilibrium state. We establish the equidistribution of weighted periodic orbits and the large deviations principle for all such measures. The thermodynamic results are proved for the class of expansive flows with weak specification.

math.DS↗
arXiv2019

Virtually cyclic dimension for 3-manifold groups

Let G be the fundamental group of a connected, closed, orientable 3-manifold. We explicitly compute its virtually cyclic geometric dimension. Among the tools we use are the prime and JSJ decompositions of M, several push-out type constructions, as well as some Bredon cohomology computations.

math.GT↗
arXiv2018

Barycentric straightening and bounded cohomology

We study the barycentric straightening of simplices in irreducible symmetric spaces of non-compact type. We show that, for an n-dimensional symmetric space of rank r>1, the p-Jacobian has uniformly bounded norm, as soon as p is at least n-r+2. As a consequence, for a non-compact, connected, semisimple real Lie group G, in degrees n-r+2 and higher, every cohomology class has a bounded representative. This answers Dupont's problem in small codimension. We also give examples of symmetric spaces where the barycentrically straightened simplices of dimension n-r have unbounded volume, showing that the range in which we obtain boundedness of the p-Jacobian is very close to optimal.

math.DG↗
arXiv2018

Primitive geodesic lengths and (almost) arithmetic progressions

In this article, we investigate when the set of primitive geodesic lengths on a Riemannian manifold have arbitrarily long arithmetic progressions. We prove that in the space of negatively curved metrics, a metric having such arithmetic progressions is quite rare. We introduce almost arithmetic progressions, a coarsification of arithmetic progressions, and prove that every negatively curved, closed Riemannian manifold has arbitrarily long almost arithmetic progressions in its primitive length spectrum. Concerning genuine arithmetic progressions, we prove that every non-compact, locally symmetric, arithmetic manifold has arbitrarily long arithmetic progressions in its primitive length spectrum. We end with a conjectural characterization of arithmeticity in terms of arithmetic progressions in the primitive length spectrum. We also suggest an approach to a well known spectral rigidity problem based on the scarcity of manifolds with arithmetic progressions.

math.DG↗
arXiv2018

Steenrod problem and the domination relation

We indicate how to combine some classical topology (Thom's work on the Steenrod problem) with some modern topology (simplicial volume) to show that every map between certain manifolds must have degree zero. We furthermore discuss a homotopy theoretic interpretation of parts of our proof, using Thom spaces and Steenrod powers.

math.AT↗
arXiv2018

Equivariant $K$-homology for hyperbolic reflection groups

We compute the equivariant $K$-homology of the classifying space for proper actions, for compact 3-dimensional hyperbolic reflection groups. This coincides with the topological $K$-theory of the reduced $C^\ast$-algebra associated to the group, via the Baum-Connes conjecture. We show that, for any such reflection group, the associated $K$-theory groups are torsion-free. As a result we can promote previous rational computations to integral compu- tations. Our proof relies on a new efficient algebraic criterion for checking torsion-freeness of K-theory groups, which could be applied to many other classes of groups.

math.KT↗
arXiv2018

Finiteness of Maximal Geodesic Submanifolds in Hyperbolic Hybrids

We show that large classes of non-arithmetic hyperbolic $n$-manifolds, including the hybrids introduced by Gromov and Piatetski-Shapiro and many of their generalizations, have only finitely many finite-volume immersed totally geodesic hypersurfaces. In higher codimension, we prove finiteness for geodesic submanifolds of dimension at least $2$ that are maximal, i.e., not properly contained in a proper geodesic submanifold of the ambient $n$-manifold. The proof is a mix of structure theory for arithmetic groups, dynamics, and geometry in negative curvature.

math.GT↗
arXiv2017

Marked length rigidity for Fuchsian buildings

We consider finite 2-complexes X that arise as quotients of Fuchsian buildings by subgroups of the combinatorial automorphism group, which we assume act freely and cocompactly. We show that locally CAT(-1) metrics on X which are piecewise hyperbolic and satisfy a natural non-singularity condition at vertices are marked length spectrum rigid within certain classes of negatively curved, piecewise Riemannian metrics on X. As a key step in our proof, we show that the marked length spectrum function for such metrics determines the volume of X.

math.MG↗
arXiv2017

Some Kähler structures on products of 2-spheres

We consider a family of Kähler structures on products of 2-spheres, arising from complex Bott manifolds. These are obtained via iterated $\mathbb P^1$-bundle constructions, generalizing the classical Hirzebruch surfaces. We show that the resulting Kähler structures all have identical Chern classes. We construct Bott diagrams, which are rooted forests with an edge labelling by positive integers, and show that these classify these Kähler structures up to biholomorphism.

math.DG↗
arXiv2016

Vanishing simplicial volume for certain affine manifolds

We show that closed aspherical manifolds supporting an affine structure, whose holonomy map is injective and contains a pure translation, must have vanishing simplicial volume. This provides some further evidence for the veracity of the Auslander Conjecture. Along the way, we provide a simple cohomological criterion for aspherical manifolds with normal amenable subgroups of $π_1$ to have vanishing simplicial volume. This answers a special case of a question due to Lück.

math.GT↗
arXiv2016

Aspherical products which do not support Anosov diffeomorphisms

We show that the product of infranilmanifolds with certain aspherical closed manifolds do not support Anosov diffeomorphisms. As a special case, we obtain that products of a nilmanifold and negatively curved manifolds of dimension at least 3 do not support Anosov diffeomorphisms.

math.DS↗