arXiv subjects

Jean-François Lafont

Publications and source records attributed to Jean-François Lafont.

24 records · Page 2Linked to original sources

arXiv2015

Combinatorial systolic inequalities

We establish combinatorial versions of various classical systolic inequalities. For a smooth triangulation of a closed smooth manifold, the minimal number of edges in a homotopically non-trivial loop contained in the $1$-skeleton gives an integer called the combinatorial systole. The number of top-dimensional simplices in the triangulation gives another integer called the combinatorial volume. We show that a class of smooth manifolds satisfies a systolic inequality for all Riemannian metrics if and only if it satisfies a corresponding combinatorial systolic inequality for all smooth triangulations. Along the way, we show that any closed Riemannian manifold has a smooth triangulation which "remembers" the geometry of the Riemannian metric, and conversely, that every smooth triangulation gives rise to Riemannian metrics which encode the combinatorics of the triangulation. We give a few applications of these results.

math.GT↗

arXiv2015

Hyperbolic groups with boundary an n-dimensional Sierpinski space

For n>6, we show that if G is a torsion-free hyperbolic group whose visual boundary is an (n-2)-dimensional Sierpinski space, then G=π_1(W) for some aspherical n-manifold W with nonempty boundary. Concerning the converse, we construct, for each n>3, examples of aspherical manifolds with boundary, whose fundamental group G is hyperbolic, but with visual boundary not homeomorphic to an (n-2)-dimensional Sierpinski space.

math.GT↗

arXiv2014

Revisiting Farrell's nonfiniteness of Nil

We study Farrell Nil-groups associated to a finite order automorphism of a ring $R$. We show that any such Farrell Nil-group is either trivial, or infinitely generated (as an abelian group). Building on this first result, we then show that any finite group that occurs in such a Farrell Nil-group occurs with infinite multiplicity. If the original finite group is a direct summand, then the countably infinite sum of the finite subgroup also appears as a direct summand. We use this to deduce a structure theorem for countable Farrell Nil-groups with finite exponent. Finally, as an application, we show that if $V$ is any virtually cyclic group, then the associated Farrell or Waldhausen Nil-groups can always be expressed as a countably infinite sum of copies of a finite group, provided they have finite exponent (which is always the case in dimension $0$).

math.KT↗

arXiv2013

Aspherical manifolds that cannot be triangulated

Although Kirby and Siebenmann showed that there are manifolds that do not admit PL structures, the possibility remained that all manifolds could be triangulated. In the late seventies Galewski and Stern and independently Matumoto showed that non-triangulable manifolds exist in all dimensions > 4 if and only if homology 3-spheres with certain properties do not exist. In 2013 Manolescu showed that, indeed, there were no such homology 3-spheres and hence, non-triangulable manifolds exist in each dimension >4. It follows from work of Freedman in 1982 that there are 4-manifolds that cannot be triangulated. In 1991 Davis and Januszkiewicz applied a hyperbolization procedure to Freedman's 4-manifolds to get closed aspherical 4-manifolds that cannot be triangulated. In this paper we apply hyperbolization techniques to the Galewski-Stern manifolds to show that there exist closed aspherical n-manifolds that cannot be triangulated for each n> 5. The question remains open in dimension 5.

math.GT↗

arXiv2012

Marked length rigidity for one dimensional spaces

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.

math.MG↗

arXiv2011

Rational equivariant K-homology of low dimensional groups

We consider groups G which have a cocompact, 3-manifold model for the classifying space \underline{E}G. We provide an algorithm for computing the rationalized equivariant K-homology of \underline{E}G. Under the additional hypothesis that the quotient 3-orbifold \underline{E}G/G is geometrizable, the rationalized K-homology groups coincide with the rationalized K-theory of the reduced C*-algebra of G. We illustrate our algorithm on some concrete examples.

math.KT↗