Related papers: A combinatorial negative curvature condition imply…

We review and extend the recently proposed model of combinatorial quantum gravity. Contrary to previous discrete approaches, this model is defined on (regular) random graphs and is driven by a purely combinatorial version of Ricci…

We study the subgroup structure of discrete groups which share cohomological properties which resemble non-negative curvature. Examples include all Gromov hyperbolic groups. We provide strong restrictions on the possible s-normal subgroups…

We show that a suitably chosen position-momentum commutator can elegantly describe many features of gravity, including the IR/UV correspondence and dimensional reduction (`holography'). Using the most simplistic example based on dimensional…

We address Gromov's band width inequality and Rosenberg's $S^1$-stability conjecture for simply connected smooth four manifolds. Both results are known to be false in dimension 4 due to counterexamples based on Seiberg-Witten invariants.…

We give a combinatorial/geometric argument of the classical result that an affine connection, which is both torsion free and curvature free, is locally an affine space.

For the $n$-body problem in spaces of negative constant Gaussian curvature, we prove for a class of negative hyperbolic rotopulsators that if that class exists, the configurations of the point masses of these rotopulsators have to be…

Two locally positive expressions for the gravitational Hamiltonian, one using 4-spinors the other special orthonormal frames, are reviewed. A new quadratic 3-spinor-curvature identity is used to obtain another positive expression for the…

We demonstrate how some problems arising in simplicial quantum gravity can be successfully addressed within the framework of combinatorial group theory. In particular, we argue that the number of simplicial 3-manifolds having a fixed…

We present four generalized small cancellation conditions for finite presentations and solve the word- and conjugacy problem in each case. Our conditions $W$ and $W^*$ contain the non-metric small cancellation cases C(6), C(4)T(4), C(3)T(6)…

In this article, we are interested in metric spaces that satisfy a weak non-positive curvature condition in the sense that they admit a conical geodesic bicombing. We show that the analog of a question of Gromov about compactness properties…

We introduce a new combinatorial condition that characterises the amenability for locally compact groups. Our condition is weaker than the well-known F{\o}lner's conditions, and so is potentially useful as a criteria to show the amenability…

We propose a computation of curvature of arbitrary two-dimensional surfaces of three-dimensional objects, which is a contribution to discrete gravity with potential applications in network geometry. We begin by linking each point of the…

We show that groups with a mild form of non-positive curvature (a navigable path system) satisfy the weak rank rigidity conjecture: they either have linear divergence or a Morse element. This class includes discrete groups of projective…

We prove that simply connected local 2-dimensional simplicial complexes embed in 3-space if and only if their dual matroids are graphic. Examples are provided that the assumptions of simply connectedness and locality are necessary. This may…

In graph theory there are intimate connections between the expansion properties of a graph and the spectrum of its Laplacian. In this paper we define a notion of combinatorial expansion for simplicial complexes of general dimension, and…

We show that the figure eight knot complement admits a uniformizable spherical CR structure, i.e. it occurs as the manifold at infinity of a complex hyperbolic orbifold. The uniformization is unique provided we require the peripheral…

We describe a new method of constructing Kobayashi-hyperbolic surfaces in complex projective 3-space based on deforming surfaces with a "hyperbolic non-percolation" property. We use this method to show that general small deformations of…

Here we extend the notion of target-local Gromov convergence of pseudoholomorphic curves to the case in which the target manifold is not compact, but rather is exhausted by compact neighborhoods. Under the assumption that the curves in…

We introduce and investigate bucolic complexes, a common generalization of systolic complexes and of CAT(0) cubical complexes. They are defined as simply connected prism complexes satisfying some local combinatorial conditions. We study…

A consequence of the surgery theorem of Gromov and Lawson is that every closed, simply-connected 6-manifold admits a Riemannian metric of positive scalar curvature. For metrics of positive Ricci curvature it is widely open whether a similar…