From symmetric spaces to buildings, curve complexes and outer spaces.
outer automorphism groups of free groups
Given a symmetric space \(X=G/K\) of noncompact type, with \(G\) the connected component of the isometry group of \(X\), the boundary of the geodesic compactification of \(X\) has the structure of a spherical Tits building, whose simplices are indexed by the proper parabolic subgroups of \(G\). The bulk of the paper under review is a survey of this well-known relation, various of its generalizations, and important applications. The unifying element is that in various settings, the structure at infinity of a noncompact space is related to a natural simplicial complex.NEWLINENEWLINE Among other things, the author discusses the Borel-Serre partial compactifications of noncompact symmetric spaces, as well as of the Teichmüller space \(T_g\) of a closed, orientable surface of genus \(\geq 2\), and of the outer space \(X_n\) of marked metric graphs with fundamental group equal to the free group of \(n\) generators. New results are presented in the last section of the paper: the author constructs a new simplicial complex, called the core graph complex, and discusses its relation to the geometry at infinity of \(X_n\).
scientific article; zbMATH DE number 6077136