The boundary of bordified 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 $C{V}_n$ on which $Out(F_n)$ acts properly and cocompactly, and is homeomorphic to the Bestvina-Feighn bordification of $C{V}_n$. In the current paper we analyze the structure of the boundary of ${\mathcal{J}}_n$. We then use the desctiption of the simplicial closure $C{V}_n^{\ast }$ 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 $C{V}_n^{\ast }$ spanned by vertices at infinity.

Typos corrected, references added to recent literature. To appear in Proceedings of TIFR International Colloquium on Randomness, Geometry and Dynamics, 1-12 January 2024

Cited by in corpus (1)

• Cohen--Macaulay Complexes, Duality Groups, and the dualizing module of $\mathrm{Out}(F_N)$