On the Filling Invariants at Infinity of Hadamard Manifolds

We study the filling invariants at infinity div _k for Hadamard manifolds defined by Brady and Farb [ Trans. Am. Math. Soc. 350(8) (1998), 3393–3405]. Among other results, we give a positive answer to the question they posed: whether these invariants can be used to detect the rank of a symmetric space of noncompact type.     

Commensurability of Hyperbolic Manifolds with Geodesic Boundary

Suppose , let M ₁, M ₂ be n-dimensional connected complete finite-volume hyperbolic manifolds with nonempty geodesic boundary, and suppose that π₁ (M ₁) is quasi-isometric to π₁ (M ₂) (with respect to the word metric). Also suppose that if n=3, then ∂M ₁ and ∂M ₂ are compact. We show that M ₁ is commensurable with M ₂. Moreover, we show that there exist homotopically equivalent hyperbolic 3-manifolds with non-compact geodesic boundary which are not commensurable with each other. We also prove that if M is as M ₁ above and G is a finitely generated group which is quasi-isometric to π₁ (M), then there exists a hyperbolic manifold with geodesic boundary M′ with the following properties: M′ is commensurable with M, and G is a finite extension of a group which contains π₁ (M′) as a finite-index subgroupMathematics Subject Classification (2000). Primary: 20F65; secondary: 30C65, 57N16     

Homological Invariants and Quasi-Isometry

Building upon work of Y. Shalom we give a homological-algebra flavored definition of an induction map in group homology associated to a topological coupling. As an application we obtain that the cohomological dimension cd_R over a commutative ring R satisfies the inequality if Λ embeds uniformly into Γ and holds. Another consequence of our results is that the Hirsch ranks of quasi-isometric solvable groups coincide. Further, it is shown that the real cohomology rings of quasi-isometric nilpotent groups are isomorphic as graded rings. On the analytic side, we apply the induction technique to Novikov-Shubin invariants of amenable groups, which can be seen as homological invariants, and show their invariance under quasi-isometry. Received: November 2004 Revision: April 2004 Accepted: April 2004     

Rigidity of quasi-isometries for some hyperbolic buildings

For a family of negatively-curved buildings, we show that any quasi-isometry lies within bounded distance from an isometry. Received: February 2, 2000.     

Instability of the Liouville property for quasi-isometric graphs and manifolds of polynomial volume growth

In this note we construct two quasi-isometric graphs. One admits an infinite dimensional space of nonconstant bounded harmonic functions, while the other admits only constant bounded harmonic functions. Translation of the construction to manifolds answers a problem due to T. Lyons.     

On homeomorphisms and quasi-isometries of the real line

We show that the group of piecewise-linear homeomorphisms of having bounded slopes surjects onto the group of all quasi-isometries of . We prove that the following groups can be imbedded in : the group of compactly supported piecewise-linear homeomorphisms of , the Richard Thompson group , and the free group of continuous rank.

     

The Tits boundary of a 2-complex

We investigate the Tits boundary of -complexes that have only a finite number of isometry types of cells. In particular, we show that away from the endpoints, a geodesic segment in the Tits boundary is the ideal boundary of an isometrically embedded Euclidean sector. As applications, we provide sufficient conditions for two points in the Tits boundary to be the endpoints of a geodesic in the -complex and for a group generated by two hyperbolic isometries to contain a free group. We also show that if two -complexes are quasi-isometric, then the cores of their Tits boundaries are bi-Lipschitz.

     

A Conjecture Concerning a Limit of Non-Cayley Graphs

Our aim in this note is to present a transitive graph that we conjecture is not quasi-isometric to any Cayley graph. No such graph is currently known. Our graph arises both as an abstract limit in a suitable space of graphs and in a concrete way as a subset of a product of trees.     

Almost Isometries of Balls

Let f be a bi-Lipschitz mapping of the Euclidean ball B ^_n into ℓ₂ with both Lipschitz constants close to one. We investigate the shape of f(B ^_n). We give examples of such a mapping f, which has the Lipschitz constants arbitrarily close to one and at the same time has in the supremum norm the distance at least one from every isometry of ⁿ.     

Classes of Operators Similar to Partial Isometries

The present paper deals with operators similar to partial isometries. We get some (necessary and) sufficient conditions for the similarity to (adjoints of) quasinormal partial isometries, or more general, to power partial isometries. We illustrate our results on the class of n-quasi-isometries, obtaining that a n-quasi-isometry is similar to a power partial isometry if and only if the ranges are closed. In particular if n = 2, these conditions ensure the similarity to quasinormal partial isometries of Duggal and Aluthge transforms of 2-quasi-isometries. The case when a n-quasi-isometry is a partial isometry is also studied, and a structure theorem for n-quasi-isometries which are power partial isometries is given. The second author was partially supported by Romanian 2-CEX Research grant. no. 06-11-34/2006.