Embeddings of Trees, Cantor Sets and Solvable Baumslag-Solitar Groups

TL;DR

This paper characterizes when quasiisometric embeddings exist between solvable Baumslag-Solitar groups and treebolic spaces, linking these to embeddings of Cantor sets and rooted trees, and confirming a conjecture of Woess.

Contribution

It extends previous work on quasiisometries by providing a comprehensive characterization of embeddings between these complex geometric structures.

Findings

Existence of embeddings is determined by boundedness of specific integer sequences.

Confirmed a conjecture of Woess regarding quasiisometries of treebolic spaces.

Established equivalences between embeddings of different mathematical objects.

Abstract

We characterise when there exists a quasiisometric embedding between two solvable Baumslag-Solitar groups. This extends the work of Farb and Mosher on quasiisometries between the same groups. More generally, we characterise when there can exist a quasiisometric embedding between two treebolic spaces. This allows us to determine when two treebolic spaces are quasiisometric, confirming a conjecture of Woess. The question of whether there exists a quasiisometric embedding between two treebolic spaces turns out to be equivalent to the question of whether there exists a bilipschitz embedding between two symbolic Cantor sets, which in turn is equivalent to the question of whether there exists a rough isometric embedding between two regular rooted trees. Hence we answer all three of these questions simultaneously. It turns out that the existence of such embeddings is completely determined by the boundedness of an intriguing family of integer sequences.