# A coarse Cartan-Hadamard theorem with application to the coarse Baum-Connes conjecture

## TL;DR

This paper proves a coarse version of the Cartan-Hadamard theorem, showing that proper coarsely convex spaces are coarsely homotopy equivalent to open cones of their boundaries, and applies this to verify the coarse Baum-Connes conjecture for these spaces and related groups.

## Contribution

It introduces a coarse Cartan-Hadamard theorem and demonstrates its application to the coarse Baum-Connes conjecture for coarsely convex spaces and systolic groups.

## Key findings

- Proper coarsely convex spaces are coarsely homotopy equivalent to open cones of their boundaries.
- Such spaces satisfy the coarse Baum-Connes conjecture.
- Systolic groups and complexes satisfy the coarse Baum-Connes conjecture.

## Abstract

We establish a coarse version of the Cartan-Hadamard theorem, which states that proper coarsely convex spaces are coarsely homotopy equivalent to the open cones of their ideal boundaries. As an application, we show that such spaces satisfy the coarse Baum-Connes conjecture. Combined with the result of Osajda-Przytycki, it implies that systolic groups and locally finite systolic complexes satisfy the coarse Baum-Connes conjecture.