The geometry of dynamical triangulations

We discuss the geometry of dynamical triangulations associated with 3-dimensional and 4-dimensional simplicial quantum gravity. We provide analytical expressions for the canonical partition function in both cases, and study its large volume behavior. In the space of the coupling constants of the theory, we characterize the infinite volume line and the associated critical points. The results of this analysis are found to be in excellent agreement with the MonteCarlo simulations of simplicial quantum gravity. In particular, we provide an analytical proof that simply-connected dynamically triangulated 4-manifolds undergo a higher order phase transition at a value of the inverse gravitational coupling given by 1.387, and that the nature of this transition can be concealed by a bystable behavior. A similar analysis in the 3-dimensional case characterizes a value of the critical coupling (3.845) at which hysteresis effects are present.Comment: 166 pages, Revtex (latex) fil

Eight lectures on Oka manifolds

Over the past decade, the class of Oka manifolds has emerged from Gromov's seminal work on the Oka principle. Roughly speaking, Oka manifolds are complex manifolds that are the target of "many" holomorphic maps from affine spaces. They are "dual" to Stein manifolds and "opposite" to Kobayashi-hyperbolic manifolds. The prototypical examples are complex homogeneous spaces, but there are many other examples: there are many ways to construct new Oka manifolds from old. The class of Oka manifolds has good formal properties, partly explained by a close connection with abstract homotopy theory. These notes were prepared for lectures given at the Institute of Mathematics of the Chinese Academy of Sciences in Beijing in May 2014. They are meant to give an accessible introduction, not to all of Oka theory, but more specifically to Oka manifolds, how they arise and what we know about them.Comment: A few improvements in version

Rough index theory on spaces of polynomial growth and contractibility

We will show that for a polynomially contractible manifold of bounded geometry and of polynomial volume growth every coarse and rough cohomology class pairs continuously with the K-theory of the uniform Roe algebra. As an application we will discuss non-vanishing of rough index classes of Dirac operators over such manifolds, and we will furthermore get higher-codimensional index obstructions to metrics of positive scalar curvature on closed manifolds with virtually nilpotent fundamental groups. We will give a computation of the homology of (a dense, smooth subalgebra of) the uniform Roe algebra of manifolds of polynomial volume growth.Comment: v4: final version, to appear in J. Noncommut. Geom. v3: added a computation of the homology of (a smooth subalgebra of) the uniform Roe algebra. v2: added as corollaries to the main theorem the multi-partitioned manifold index theorem and the higher-codimensional index obstructions against psc-metrics, added a proof of the strong Novikov conjecture for virtually nilpotent groups, changed the titl

Entropy of random coverings and 4D quantum gravity

We discuss the counting of minimal geodesic ball coverings of $n$-dimensional riemannian manifolds of bounded geometry, fixed Euler characteristic and Reidemeister torsion in a given representation of the fundamental group. This counting bears relevance to the analysis of the continuum limit of discrete models of quantum gravity. We establish the conditions under which the number of coverings grows exponentially with the volume, thus allowing for the search of a continuum limit of the corresponding discretized models. The resulting entropy estimates depend on representations of the fundamental group of the manifold through the corresponding Reidemeister torsion. We discuss the sum over inequivalent representations both in the two-dimensional and in the four-dimensional case. Explicit entropy functions as well as significant bounds on the associated critical exponents are obtained in both cases.Comment: 54 pages, latex, no figure

Thurston obstructions and Ahlfors regular conformal dimension

Let $f: S^2 \to S^2$ be an expanding branched covering map of the sphere to itself with finite postcritical set $P_f$. Associated to $f$ is a canonical quasisymmetry class \GGG(f) of Ahlfors regular metrics on the sphere in which the dynamics is (non-classically) conformal. We show \inf_{X \in \GGG(f)} \hdim(X) \geq Q(f)=\inf_\Gamma \{Q \geq 2: \lambda(f_{\Gamma,Q}) \geq 1\}. The infimum is over all multicurves $\Gamma \subset S^2-P_f$. The map $f_{\Gamma,Q}: \R^\Gamma \to \R^\Gamma$ is defined by $$f_{\Gamma, Q}(\gamma) =\sum_{[\gamma']\in\Gamma} \sum_{\delta \sim \gamma'} \deg(f:\delta \to \gamma)^{1-Q}[\gamma'],$$ where the second sum is over all preimages $\delta$ of $\gamma$ freely homotopic to $\gamma'$ in $S^2-P_f$, and $\lambda(f_{\Gamma,Q})$ is its Perron-Frobenius leading eigenvalue. This generalizes Thurston's observation that if $Q(f)>2$, then there is no $f$-invariant classical conformal structure.Comment: Minor revisions are mad