# Jean-François Lafont

E-mail: jlafont@math.ohio-state.edu
Office: MW 426
Phone: (614)-292-4010

## Other Information:

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

## Editorial Work:

I am a member of the Editorial Board for the London Mathematical Society Student Texts book series.
Editor for the proceedings for the ICM satellite conference "Geometry, topology and dynamics in negative curvature". Published by the London Mathematical Society, Lecture Notes series, Vol. 425.
Editor for the proceedings for the OSU special year "Topology and geometric group theory". Published by Springer-Verlag, Proceedings in Mathematics series, Vol. 184.
Editor for the proceedings for the conference "Topological methods in group theory", held in honor of Ross Geoghegan. Published by the London Mathematical Society, Lecture Notes series, Vol. 451.

## Former Students:

Raeyong Kim (2012) - On the theorem of Kan-Thurston and algebraic rank of CAT(0) groups. (Co-advised by Ian Leary)
Kyle Joecken (2013) - Dimension of classifying spaces for virtually cyclic subgroups of certain geometric groups.
Ryan Kowalick (2013) - Discrete systolic inequalities.
Kun Wang (2014) - On the Farrell-Jones conjecture.
Andy Nicol (2014) - Quasi-isometries of graph manifolds do not preserve non-positive curvature.
Shi Wang (2016) - Barycentric straightening, splitting rank and bounded cohomology.
Bakul Sathaye (2018) - Obstructions to Riemannian smoothings of locally CAT(0) manifolds.
Chris Kennedy (2018) - Construction of maps by Postnikov towers.

## 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), DMS-1510640 (2015-2018), DMS-1812028 (2018-2021) 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, M. Bucher, C. Connell, D. Constantine, M. Davis, F.T. Farrell, D. Fisher, J. Fowler, S. Francaviglia, R. Frigerio, A. Gogolev, T. Januszkiewicz, D. Juan-Pineda, A. Kar, R. Kowalick, B. Magurn, D.B. McReynolds, S. Millan-Vossler, N. Miller, A. Minasyan, B. Minemyer, C. Neofytidis. I. Ortiz, S. Pallekonda, Ch. Pittet, E. Prassidis, A. Rahm, R. Roy, R. Sánchez-García, B. Schmidt, A. Sisto, G. Sorcar, M. Stover, D. Thompson, B. Tshishiku, W. van Limbeek, Kun Wang, Shi Wang, and Fangyang Zheng.

## Submitted papers:

• Strong symbolic dynamics for geodesic flow on CAT(-1) spaces and other metric Anosov flows (joint with D. Constantine and D. Thompson)

Abstract. 25 pages as a preprint (Aug. 2018)

We prove that the geodesic flow on a compact, locally CAT(-1) metric space 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.

• Steenrod problem and the domination relation (joint with C. Neofytidis)

Abstract. 9 pages as a preprint (Aug. 2018)

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.

• Filling triangulated surfaces (joint with R. Kowalick and B. Minemyer)

Abstract. 14 pages as a preprint (Apr. 2018)

Given a triangulated closed oriented surface M, we provide upper bounds on the number of tetrahedra needed to construct a triangulated 3-manifold N which bounds M. Along the way, we develop a technique to translate (in all dimensions) between the famous Riemannian systolic inequalities of Gromov and combinatorial analogues of these inequalities.

• Finiteness of maximal geodesic submanifolds in hyperbolic hybrids (joint with D. Fisher, N. Miller, and M. Stover)

Abstract. 31 pages as a preprint (Feb. 2018)

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 two 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.

## Accepted papers:

• The weak specification property for geodesic flows on CAT(-1) spaces (joint with D. Constantine and D. Thompson)

Abstract. 33 pages as a preprint (June 2016). To appear in Groups Geom. Dyn.

We prove that the geodesic flow on a compact locally CAT(-1) space has the weak specification property, and give various applications of this property. We show that every Holder continuous function on the space of geodesics has a unique equilibrium state, and as a result, that the Bowen-Margulis measure is the unique measure of maximal entropy. We establish the equidistribution of weighted periodic orbits and the large deviations principle for all such measures. For compact locally CAT(0) spaces, we give partial results, both positive and negative, on the specification property and the existence of a coding of the geodesic flow by a suspension flow over a compact shift of finite type.

• Some Kähler structures on products of 2-spheres (joint with G. Sorcar and F. Zheng)

Abstract. 14 pages as a preprint (Aug. 2017). To appear in Enseign. Math.

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.

• Equivariant K-homology for hyperbolic reflection groups (joint with I. Ortiz, A. Rahm, and R. Sánchez-García)

Abstract. 29 pages as a preprint (July 2017). To appear in Q. J. Math.

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 $$C^*$$-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. This means that previous rational computations can now be promoted to integral computations.

[The arXived version is 27 pages longer, consisting of all the tables required to support the calculations in the main theorem. These tables will be omitted in the published version.]

• Fat flats in rank-one manifolds (joint with D. Constantine, D.B. McReynolds, and D. Thompson)

Abstract. 23 pages as a preprint (Feb. 2017). To appear in Michigan Math. J.

We study closed non-positively curved Riemannian manifolds M which admit fat k-flats: that is, the universal cover $$\tilde M$$ contains a positive radius neighborhood of a k-flat on which the sectional curvatures are identically zero. We investigate how the fat k-flats affect the cardinality of the collection of closed geodesics. Our first main result is to construct rank-one non-positively curved manifolds where the fat flat corresponds to a twisted cylindrical neighborbood of a geodesic on M. As a result, M contains an embedded periodic geodesic with a flat neighborhood, but M nevertheless has only countably many closed geodesics. Such metrics can be constructed on finite covers of arbitrary odd-dimensional finite volume hyperbolic manifolds. Our second main result is to prove a closing theorem for fat flats, which implies that a manifold M with a fat k-flat contains an immersed, totally geodesic k-dimensional flat closed submanifold. This guarantees the existence of uncountably many closed geodesics when $$k \geq 2$$. Finally, we collect results on thermodynamic formalism for the class of manifolds considered in this paper.

• Marked length rigidity for Fuchsian buildings (joint with D. Constantine)

Abstract. 29 pages as a preprint (Dec. 2017). To appear in Ergodic Theory Dynam. Systems.

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.

• Sets of degrees of maps between SU(2)-bundles over the 5-sphere (joint with C. Neofytidis)

Abstract. 8 pages as a preprint (Oct. 2017). To appear in Transform. Groups.

We compute the sets of degrees of maps between SU(2)-bundles over the 5-sphere. We show that the only obstruction to the existence of a mapping degree between those manifolds is derived by the Steenrod squares. We construct explicit maps realizing each integer that occurs as a mapping degree between these bundles.

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

Abstract. 39 pages as a preprint (Sept. 2012). To appear in J. Topol. Anal.

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.

• Primitive geodesic lengths and (almost) arithmetic progressions (joint with D.B. McReynolds)

Abstract. 29 pages as a preprint (May 2014). To appear in Publ. Mat.

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. We also prove that every noncompact arithmetic hyperbolic 2- or 3-manifold has arbitrarily long arithmetic progressions in its primitive length spectrum. We end with a conjectural characterization of arithmeticity.

• Hyperbolic groups with boundary an n-dimensional Sierpinski space (joint with B. Tshishiku)

Abstract. 13 pages as a preprint (May 2015). To appear in J. Topol. Anal.

For $$n\geq 7$$, we show that if $$G$$ is a torsion-free hyperbolic group whose visual boundary is an $$(n-2)$$-dimensional Sierpinski space, then $$G=\pi_1(W)$$ for some aspherical $$n$$-manifold W with nonempty boundary. Concerning the converse, we construct, for each $$n\geq 4$$, examples of locally CAT(-1) manifolds with non-empty totally geodesic boundary, but with visual boundary not homeomorphic to an $$(n-2)$$-dimensional Sierpinski space.

• Barycentric straightening and bounded cohomology (joint with S. Wang)

Abstract. 22 pages as a preprint (July 2015). To appear in J. Eur. Math. Soc. (JEMS)

We study the barycentric straightening of simplices in higher rank irreducible symmetric spaces of non-compact type. We show that, for an $$n$$-dimensional symmetric space of rank $$r \geq 2$$ (excluding $$SL(3, \mathbb R)/ SO(3)$$ and $$SL(4, \mathbb R)/SO(4)$$), the $$p$$-Jacobian has uniformly bounded norm, provided $$p \geq n-r+2$$. As a consequence, for the corresponding non-compact, connected, semisimple real Lie group $$G$$, in degrees $$p \geq n-r+2$$, every degree $$p$$ 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.

## Published papers:

### 2018

• Vanishing simplicial volume for certain affine manifolds (joint with M. Bucher and C. Connell)

Abstract. 8 pages as a preprint (Sept. 2016). Final version in Proc. Amer. Math. Soc. 146 (2018), pgs. 1287-1294.

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 $$\pi_1$$ to have vanishing simplicial volume. This answers a special case of a question due to Lück.

### 2017

• Quasicircle boundaries and exotic almost-isometries (joint with B. Schmidt and W. van Limbeek)

Abstract. 16 pages as a preprint (Sept. 2014). Final version in Ann. Inst. Fourier 67 (2017), pgs. 863-877.

We consider properly discontinuous, isometric, convex co-compact actions of surface groups $$\Gamma$$ on a CAT(-1) space $$X$$. We show that the limit set of such an action, equipped with the canonical visual metric, is a (weak) quasicircle in the sense of Falconer and Marsh. It follows that the visual metrics on such limit sets are classified, up to bi-Lipschitz equivalence, by their Hausdorff dimension. This result applies in particular to boundaries at infinity of the universal cover of a locally CAT(-1) surface. We show that any two periodic CAT(-1) metrics on $$\mathbb H^2$$ can be scaled so as to be almost-isometric (though in general, no equivariant almost-isometry exists). We also construct, on each higher genus surface, $$k$$-dimensional families of equal area Riemannian metrics, with the property that their lifts to the universal covers are pairwise almost-isometric but are not isometric to each other. Finally, we exhibit a gap phenomena for the optimal multiplicative constant for a quasi-isometry between periodic CAT(-1) metrics on $$\mathbb H^2$$.

### 2016

• Aspherical products which do not support Anosov diffeomorphisms (joint with A. Gogolev)

Abstract. 18 pages as a preprint. Final version in Ann. Henri Poincaré 17 (2016), pgs. 3005-3026.

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 three do not support Anosov diffeomorphisms.

Errata: The main theorem is correct as stated, but the order of some of the steps of the proof in the published paper need to be switched. Specifically, the passing to a cover in order to orient the stable and unstable subbundles (Section 6.1) should occur after the analysis of the induced automorphism (Section 6.2) and the construction of the model map (Section 6.3). This is because our group theoretic hypotheses (which are used exclusively in Sections 6.2, 6.3) are not necessarily preserved by passing to finite index subgroups. We thank C. Neofytidis for pointing out this issue to us.

• On the Hausdorff dimension of CAT(k) surfaces (joint with D. Constantine)

Abstract. 13 pages as a preprint. Final version in Anal. Geom. Metr. Spaces 4 (2016), pgs. 266-277.

We prove a that a closed surface with a CAT(k) metric has Hausdorff dimension = 2, and that there are uniform upper and lower bounds on the two-dimensional Hausdorff measure of small metric balls. We also discuss a connection between this uniformity condition and some results on the dynamics of the geodesic flow for such surfaces. Finally, we give a short proof of topological entropy rigidity for geodesic flow on certain CAT(-1) manifolds.

• Rigidity of almost-isometric universal covers (joint with A. Kar and B. Schmidt)

Abstract. 24 pages as a preprint. Final version in Indiana Univ. Math. J. 65 (2016), pgs. 585-613.

Almost-isometries are quasi-isometries with multiplicative constant one. Lifting a pair of metrics on a compact space gives quasi-isometric metrics on the universal cover. Under some additional hypotheses on the metrics, we show that there is no almost-isometry between the universal covers. We show that Riemannian manifolds which are almost-isometric have the same volume growth entropy. We establish various rigidity results as applications.

• Revisiting Farrell's nonfiniteness of Nil (joint with S. Prassidis and K. Wang)

Abstract. 17 pages as a preprint. Final version in Ann. K-theory 1 (2016), pgs. 209-225.

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).

### 2015

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

Abstract. 171 + xxii pages as a preprint. Monograph published as Astérisque Vol. 372 (2015).

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 $$\geq 3$$. Our main goal is to study this class of graph manifolds from the viewpoint of rigidity theory.

We show that, in dimensions $$\geq 6$$, 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. These form a subclass which has better coarse geometric properties, in that various subgroups can be shown to be quasi-isometrically embedded inside the fundamental group. 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. Along the way, we classify groups which are quasi-isometric to the product of a free abelian group and a non-uniform lattice in $$SO(n,1)$$. We provide various examples of graph manifolds which do not support any locally CAT(0) metric.

Several of our results can be extended to allow pieces with hyperbolic surface factors. We emphasize that, in dimension $$=3$$, our notion of graph manifold does not coincide with the classical graph manifolds. Rather, it is a class of $$3$$-manifolds that contains some (but not all) classical graph $$3$$-manifolds (we don't allow general Seifert fibered pieces), as well as some non-graph $$3$$-manifolds (we do allow hyperbolic pieces).

### 2014

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

Abstract. 13 pages as a preprint Final version in J. Group Theory 17 (2014), pgs. 361-378.

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.

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

Abstract. 10 pages as a preprint. Final version in Algebr. Geom. Topol. 14 (2014), pgs. 795-803.

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.

### 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''. ]

• Combinatorial systolic inequalities (joint with R. Kowalick and B. Minemyer)

Abstract. 30 pages as a preprint (June 2015)

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.

[A referee pointed out that several of the results in this paper were already known (though with different proofs). The portion of the paper that was not previously known has been split off into a separate paper with Ryan and Barry: Filling triangulated surfaces''. ]

## 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.]