TL;DR
This paper introduces the pants graph for free groups, exploring its properties and actions, and establishing its connectivity and unboundedness.
Contribution
It defines the pants graph for free groups, analyzes the outer automorphism group's action, and relates it to the free splitting complex.
Findings
The pants graph is connected.
The pants graph is unbounded.
The outer automorphism group acts isometrically on the pants graph.
Abstract
We introduce the concept of a pants decomposition for a finitely generated free group and construct the corresponding pants graph. A pants decomposition of a free group leads to the formation of a simplicial graph, referred to as the pants graph of a free group, consisting of all possible pants decompositions. The natural isometric action of the outer automorphism group of the free group on the pants graph induces a coarsely surjective orbit map. Additionally, we construct a coarsely Lipschitz map from the pants graph to the free splitting complex. These results imply that the pants graph of a free group is both connected and unbounded.
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