Jean-François Lafont
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?)
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.
Current Students:
Chris Kennedy,
Bakul Sathaye.
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),
and by an Alfred P. Sloan Research Fellowship (2008-2012).
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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,
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,
A. Minasyan,
B. Minemyer,
C. Neofytidis.
I. Oppenheim,
I. Ortiz,
S. Pallekonda,
Ch. Pittet,
E. Prassidis,
A. Rahm,
R. Roy,
R. Sánchez-García,
B. Schmidt,
A. Sisto,
G. Sorcar,
D. Thompson,
B. Tshishiku,
W. van Limbeek,
Kun Wang,
Shi Wang, and
Fangyang Zheng.
Submitted papers:
- 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)
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.
- Some Kähler structures on products of 2-spheres
(joint with G. Sorcar and F. Zheng)
Abstract.
12 pages as a preprint (Aug. 2017)
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.
54 pages as a preprint (July 2017)
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.
- Fat flats in rank-one manifolds
(joint with D. Constantine, D.B. McReynolds, and D. Thompson)
Abstract.
23 pages as a preprint (Feb. 2017)
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.
- 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)
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.
- 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.
Accepted papers:
- 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.
- Vanishing simplicial volume for certain affine manifolds
(joint with M. Bucher and C. Connell)
Abstract.
8 pages as a preprint (Sept. 2016). To appear in Proc. Amer. Math. Soc.
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.
- 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:
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.
- 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
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 l^{1}-seminorm and the manifold seminorm on integral homology classes.
The l^{1}-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 l^{1}-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.
- 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
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").
- 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.
- 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:
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.]
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