Publication

Isoperimetry on Finitely Generated Groups and Metric Almost Convexity Condition

2025

Abstract

We improve the isoperimetric inequality on finitely generated groups. We calculate the isoperimetric profile for homogeneous trees, accompanied by examples of optimal subsets. Additionally, we provide guidelines for constructing new optimal subsets.

We generalize the notion of almost convexity on Cayley graphs to metric spaces, introducing the concept of metric almost-convexity. Furthermore, we demonstrate that horospherical products do not satisfy the Poénaru condition.

Finally, we prove that if two geodesics metric spaces are rough-isometrics and one of them satisfies the condition of metric almost-convexity, then the other space also satisfies this condition.

Official source

https://infoscience.epfl.ch/entities/publication/f44281e2-f3a6-45b2-a43f-6f2ab4714e89

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2025

Abstract

Finally, we prove that if two geodesics metric spaces are rough-isometrics and one of them satisfies the condition of metric almost-convexity, then the other space also satisfies this condition.

Official source

https://infoscience.epfl.ch/entities/publication/f44281e2-f3a6-45b2-a43f-6f2ab4714e89

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Related concepts (32)

Metric space

In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane.

Intrinsic metric

In the mathematical study of metric spaces, one can consider the arclength of paths in the space. If two points are at a given distance from each other, it is natural to expect that one should be able to get from the first point to the second along a path whose arclength is equal to (or very close to) that distance. The distance between two points of a metric space relative to the intrinsic metric is defined as the infimum of the lengths of all paths from the first point to the second.

Complete metric space

In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). For instance, the set of rational numbers is not complete, because e.g. is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges to it (see further examples below).

Related publications (3)

Isoperimetry on Finitely Generated Groups and Metric Almost Convexity Condition

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