A local cut point is by definition a point that disconnects its sufficiently small neighborhood. We show that there exists an upper bound for the degree of a local cut point in a metric measure space satisfying the generalized Bishop–Gromov inequality. As a corollary, we obtain an upper bound for the number of ends of such a space. We also obtain some obstruction conditions for the existence of...

Let G be a countable group which acts by isometries on a separable, but not necessarily proper, Gromov hyperbolic space X. We say the action of G is weakly hyperbolic if G contains two independent hyperbolic isometries. We show that a random walk on such G converges to the Gromov boundary almost surely. We apply the convergence result to show linear progress and linear growth of translation len...

Let k and k′ be two knots in the 3-sphere. Say k 1-dominates k′, if there is a proper degree 1 map f : E(k) → E(k′). Theorem: Suppose that any companion of k is prime. If k 1-dominates k′ with the same Gromov volume, then k′ can be obtained from k by finitely many de-satellizations. The condition of “same Gromov volume” clearly cannot be removed. We also give a new construction of 1-domination ...

We present an extension of the Gromov-Hausdorff metric on the set of compact metric spaces: the Gromov-Hausdorff-Prokhorov metric on the set of compact metric spaces endowed with a finite measure. We then extend it to the non-compact case by describing a metric on the set of rooted complete locally compact length spaces endowed with a boundedly finite measure. We prove that this space with the ...

The Gromov–Hausdorff distance between metric spaces appears to be a useful tool for modeling some object matching procedures. Since its conception it has been mainly used by pure mathematicians who are interested in the topology generated by this distance, and quantitative consequences of the definition are not very common. As a result, only few lower bounds for the distance are known, and the ...

For any smooth complex projective variety X and smooth very ample hypersurface Y ⊂ X, we develop the technique of genus zero relative Gromov-Witten invariants of Y in X in algebro-geometric terms. We prove an equality of cycles in the Chow groups of the moduli spaces of relative stable maps that relates these relative invariants to the Gromov-Witten invariants of X and Y . Given the Gromov-Witt...