# Exponential growth of homotopy groups of suspended finite complexes

## TL;DR

This paper investigates the exponential growth of homotopy groups in suspended finite complexes, establishing conditions under which these spaces exhibit local hyperbolicity, with specific results for Moore spaces.

## Contribution

It introduces the concept of local hyperbolicity for spaces based on homotopy group growth and proves that suspended finite complexes are locally hyperbolic under certain conditions.

## Key findings

- Moore spaces are locally hyperbolic.
- Conditions for local hyperbolicity are related to functorial decomposition.
- Other candidate spaces are identified as locally hyperbolic.

## Abstract

We study the asymptotic behavior of the homotopy groups of simply connected finite $p$-local complexes, and define a space to be locally hyperbolic if its homotopy groups have exponential growth. Under some certain conditions related to the functorial decomposition of loop suspension, we prove that the suspended finite complexes are locally hyperbolic if suitable but accessible information of the homotopy groups is known. In particular, we prove that Moore spaces are locally hyperbolic, and other candidates are also given.