Assuming outer space has a combinatorial structure, Handel (Lehman College) and Mosher (Rutgers) consider an axis in the Culler-Vogtmann outer space Xinfr of a finite rank free group Finfr with respect to the action of a nongeometric, fully irreducible outer automorphism
Looking at recent results in the area of ergodic theory (the mathematical study of dynamical systems with an invariant measure) concerning the complexity of the problem of classification of ergodic measure preserving transformations up to conjugacy, the structure of the outer automorphism
group of a countable measure preserving equivalence relation, ergodic theoretic characterizations with the Haagerup approximation property, and cocycle superrigidity, the author of this monograph realized that these apparently diverse results can all be understood within a unified framework.
Papers cover such subjects as outer automorphism
groups of certain orientable Seifert three-manifold groups, a proposed public key cryptosystem using the modular group, normal subgroups of themodular group and other Hecke groups, unions of varieties and quasi-varieties, context-free irreducible word problems in groups, informative words and discreteness, using group theory for knowledge representation and discovery, torsion in maximal arithmetic Fuchsian groups, density of test elements in finite Abelian groups and the Rosenberg "monster.
ii) (g, h) is isomorphic to one of (C) (H) up to outer automorphisms