The boundary of bordified Outer space

TL;DR

This paper investigates the boundary structure of the Jewel space, an equivariant deformation retract of Outer space, revealing its homotopy equivalence to a subcomplex of the sphere complex associated with a connected sum of $S^1\times S^2$.

Contribution

It provides a detailed analysis of the boundary of the Jewel space and establishes its homotopy equivalence to a specific subcomplex of the sphere complex.

Findings

The boundary of the Jewel space is homotopy equivalent to a subcomplex of the sphere complex.

The boundary structure is analyzed using the description of the simplicial closure of Outer space.

The boundary relates to the Bestvina-Feighn bordification of Outer space.

Abstract

We study the boundary of the "Jewel space" $\mathcal J_n$ constructed in arXiv:1709.01296. This is an equivariant deformation retract of Outer space $CV_n$ on which $Out(F_n)$ acts properly and cocompactly, and is homeomorphic to the Bestvina-Feighn bordification of $CV_n$. In the current paper we analyze the structure of the boundary of $\mathcal J_n$. We then use the desctiption of the simplicial closure $CV_n^*$ as the sphere complex of a connected sum of $n$ copies of $S^1\times S^2$ to prove that this boundary is homotopy equivalent to the subcomplex of $CV_n^*$ spanned by vertices at infinity.