Journal of Topology and Analysis

If $(M^n,g)$ is a closed Riemannian manifold where every unit ball has volume at most ${ϵ}_n$ (a sufficiently small constant), then the $(n-1)$-dimensional Uryson width of $(M^n,g)$ is at most 1.

It is shown that the growth rate $({lim}_r|B(r){|}^{1/r})$ of any $k$ faces Dirichlet tiling of ${ℍ}^d,d>2,$ is at most $k-1-ϵ$, for an $ϵ>0$, depending only on $k$ and $d$. We do not know if there is a universal ${ϵ}_u>0$, such that $k-1-{ϵ}_u$ upperbounds the growth rate for any $k$-regular tiling, when $d>2$?

Gromov conjectured that any irreducible lattice in a symmetric space of rank at least $3$ should have at most polynomial Dehn function. We prove that the lattice $\mathrm{\text{Sp}}(2p;\mathbb{Z})$ has quadratic Dehn function when $p\ge 5$. By results of Broaddus, Farb, and Putman, this implies that the Torelli group in large genus is at most exponentially distorted.

We estimate the operator norm of radial non-negative functions on hyperbolic groups. As a consequence, we show that several forms of Haagerup’s inequality are optimal.

For bi-Hilbertian $A$-bimodules, in the sense of Kajiwara–Pinzari–Watatani, we construct a Kasparov module representing the extension class defining the Cuntz–Pimsner algebra. The construction utilises a singular expectation which is defined using the $C^{\ast }$-module version of the Jones index for bi-Hilbertian bimodules. The Jones index data also determines a novel quasi-free dynamics and KMS states on these Cuntz–Pimsner algebras.

In this paper we prove the existence of a natural mapping from the surgery exact sequence for topological manifolds to the analytic surgery exact sequence of Higson and Roe. This generalizes the fundamental result of Higson and Roe, but in the treatment given by Piazza and Schick, from smooth manifolds to topological manifolds. Crucial to our treatment is the Lipschitz signature operator of Teleman.

We also give a generalization to the equivariant setting of the product defined by Siegel in his Ph.D. thesis. Geometric applications are given to stability results for rho classes. We also obtain a proof of the APS delocalized index theorem on odd dimensional manifolds, both for the spin Dirac operator and the signature operator, thus extending to odd dimensions the results of Piazza and Schick. Consequently, we are able to discuss the mapping of the surgery sequence in all dimensions.

We construct a sequence of balanced finite presentations of the trivial group with two generators and two relators with the following property: The minimal number of relations required to demonstrate that a generator represents the trivial element grows faster than the tower of exponentials of any fixed height of the length of the finite presentation.