Block and Weinberger show that an arithmetic manifold can be endowed with a positive scalar curvature metric if and only if its ℚ-rank exceeds 2. We show in this article that these metrics are never in the same coarse class as the natural metric inherited from the base Lie group. Furthering the coarse C^*-algebraic methods of Roe, we find a nonzero Dirac obstruction in the K-theory of a particular operator algebra which encodes information about the quasi-isometry type of the manifold as well as its local geometry.

We give the first explicit lower bound for the length of a geodesic in a closed orientable hyperbolic 3-manifold M of lowest volume. We also give an upper bound for the tube radius of any shortest geodesic in M. We explain how these results might be the first steps towards a rigorous computer assisted effort to determine the least volume closed orientable hyperbolic 3-manifold(s).

Let $N$ be a hyperbolic 3-manifold and $B$ a component of the interior of $AH(\pi_1(N))$, the space of marked hyperbolic 3-manifolds homotopy equivalent to $N$. We will give topological conditions on $N$ sufficient to give $\mathcal{p}\in \overline{B}$ such that for every sufficiently small neighbourhood $V$ of $\mathcal{p}, V \bigcap B$ is disconnected. This implies that $\overline{B}$ is not a manifold with boundary.

In this paper, we describe great details of the bubbling behavior for a sequence of solutions $\mathcal{w}_i$ of $$L\mathcal{w}_1 + R_i)\mathcal{w}_i^{\frac {n+2}{n-2}} = 0 \,\, \rm{on}\, S^n, $$ where $L$ is the conformal Laplacian operator of $(S^n, g_0)$ and $R_i = n(n-2) + t_i, \hat{R}, \hat{R} \in C^1(S^n)$. As $t_i \downarrow 0$ we prove among other things the location of blowup points, the spherical Harnack inequality near each blowup point and the asymptotic formulas for the interaction of different blowup points. This is the first step toward computing the topological degree for the nonlinear PDE.

We study the action of the fundamental group $\Gamma$ of a negatively curved 3-manifold $M$ on the universal cover $\tilde{M}$ of $M$. In particular we consider the ergodicity properties of the action and the distances by which points of $\tilde{M}$ are displaced by elements of $\Gamma$. First we prove a displacement estimate for a general $n$-dimensional manifold with negatively pinched curvature and free fundamental group. This estimate is given in terms of the critical exponent $D$ of the Poincaré series for $\Gamma$. For the case in which $n= 3$, assuming that $\Gamma$ is free of rank $k \geq 2$, that the limit set of $\Gamma$ has positive 2-dimensional Hausdorff measure, that $D = 2$ and that the Poincaré series diverges at the exponent 2, we prove a displacement estimate for $\Gamma$ which is identical to the one given by the log$(2k-1)$ theorem [1] for the constant-curvature case.