Bi-Lipschitz geometry of quasiconformal trees

Bi-Lipschitz geometry of quasiconformal trees

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Vyron Vellis, University of Tennessee
Fine Hall 314

In-Person Talk

A quasiconformal tree is a doubling metric tree in which the diameter of each arc is bounded above by a fixed multiple of the distance between its endpoints. In this talk, we are concerned with two questions related to the bi-Lipschitz geometry of these metric spaces; uniformization and embeddability. First, we show that every quasiconformal tree bi-Lipschitz embeds in a Euclidean space. Second, we construct a catalog of metric trees in a purely combinatorial way, and show that every quasiconformal tree is bi-Lipschitz equivalent to one of the trees in our catalog. Finally, we discuss how such constructions apply to a special class of bounded turning and doubling metric spaces. This is inspired by results of Herron-Meyer and Rohde for quasi-arcs.

The talk is based on joint works with G. C. David and S. Eriksson-Bique.