Finite Quotient of Join in Alexandrov Geometry

TL;DR

This paper proves that certain Alexandrov spaces with curvature ≥ 1 can be represented as finite quotients of joins of convex subsets, extending the understanding of their geometric structure.

Contribution

It establishes conditions under which an Alexandrov space is isometric to a finite quotient of a join of convex subsets, generalizing join decompositions.

Findings

Spaces with specified convex subsets are finite quotients of joins.

Conditions for isometry to join quotients are characterized.

Provides a structural decomposition for Alexandrov spaces with curvature ≥ 1.

Abstract

Given two $n_i$-dimensional Alexandrov spaces $X_i$ of curvature $\ge 1$, the join of $X_1$ and $X_2$ is an $(n_1+n_2+1)$-dimensional Alexandrov space $X$ of curvature $\ge 1$, which contains $X_i$ as convex subsets such that their points are $\frac \pi2$ apart. If a group acts isometrically on a join that preserves $X_i$, then the orbit space is called quotient of join. We show that an $n$-dimensional Alexandrov space $X$ with curvature $\ge 1$ is isometric to a finite quotient of join, if $X$ contains two compact convex subsets $X_i$ without boundary such that $X_1$ and $X_2$ are at least $\frac \pi2$ apart and $\dim(X_1)+\dim(X_2)=n-1$.