Random trees found in Outer space boundary.
problem Understanding trees in the boundary of Outer space.
method Proving properties of harmonic measure on random walks.
result Typical tree is trivalent and nongeometric.
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We prove that a "random" free group outer automorphism is an ageometric fully irreducible outer automorphism whose ideal Whitehead graph is a union of triangles. In particular, we show that its attracting (and repelling) tree is a nongeometric -tree all of whose branch points are trivalent
In this paper, we study a flag complex which is naturally associated to the Thurston theory of surface diffeomorphisms for compact connected orientable surfaces with boundary. The various pieces of the Thurston decomposition of a surface diffeomorphism, thick domains and annular or thin domains, fit into this flag comp…
New braid group representations into mapping class groups are derived from disk coverings.
problem Tackling the representation of braid groups into mapping class groups.
method Explicit expressions of braid group representations into automorphism groups of free groups using covering groupoids.
result Each generator of braid group inside mapping class group is a product of Dehn twists.
Decomposes axis bundles into cubist structures for fully irreducible outer automorphisms.
problem Understanding the geometry and structure of axis bundles for fully irreducible outer automorphisms.
method Develops a 'cubist' decomposition into branched cubes with special combinatorics.
result Locates a canonical finite collection of periodic fold lines in each axis bundle.
Framework for reconstructing knot exterior surface complexes.
problem Understanding the mapping class group action on knot exteriors.
method Organizing rigidity questions, using a reduction principle, and classifying automorphisms.
result Common image-kernel bookkeeping and reconstruction framework for knot exteriors.