Abstract
The self-similar structure of the attracting subshift of a primitive substitution is carried over to the limit set of the repelling tree in the boundary of outer space of the corresponding irreducible outer automorphism of a free group. Thus, this repelling tree is self-similar (in the sense of graph directed constructions). Its Hausdorff dimension is computed. This reveals the fractal nature of the attracting tree in the boundary of outer space of an irreducible outer automorphism of a free group.
Highlights
The self-similar structure of the attracting subshift of a primitive substitution is carried over to the limit set of the repelling tree in the boundary of Outer Space of the corresponding irreducible outer automorphism of a free group
Vogtmann’s Outer Space CVN, such that the Hausdorff dimension of their metric completion T is strictly bigger than 1
It is proven in [BFH97] that the attracting lamination does only depend on the irreducible outer automorphism Φ and not on the choice of the train-track representative τ
Summary
The free group FN is Gromov-hyperbolic and has a well defined boundary at infinity ∂FN , which is a topological space, a Cantor set. The action of FN on its boundary is by homeomorphisms. A lamination (in its algebraic setting) is a closed, FN -invariant, flipinvariant subset of ∂2FN (where the flip is the map exchanging the two coordinates of a line). The elements of a lamination are called leaves. We refer the reader to [CHL08a] where laminations for free groups are defined and different equivalent approaches are exposed with care
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