Year: 2024
Type: article
Abstract: We introduce the notion of mixed subtree quasi-isometries, which are self-quasi-isometries of regular trees built in a specific inductive way.We then show that any self-quasi-isometry of a regular tree is at bounded distance from a mixed-subtree quasi-isometry.Since the free group is quasi-isometric to a regular tree, this provides a way to describe all self-quasi-isometries of the free group.In doing this, we also give a way of constructing quasi-isometries of the free group. 20F65 2 PreliminariesIn this section, we introduce the relevant notation and some preliminary lemmas.Throughout this paper, we will view F 2 as a rooted tree.Therefore, our results will cover self-quasi-isometries of rooted trees.Definition 2.1 Let .X; d / be a metric space, we say that a map f W X !X is a C -quasi-isometric embedding for a constant C 1 if d.x; y/ C C Ä d.f .x/;f .y//Ä Cd.x; y/ C C for all x; y 2 X.Further, we say that a C -quasi-isometric embedding f W X !X is a C -quasi-isometry if there exists a constant D such that for all y 2 X there exists x 2 X such that d.y; f .x//Ä D.Definition 2.2 Let .X; d / be a metric space.Two maps f; g W X !X are C -bounded if d.f .x/;g.x// Ä C for all x 2 X.They are bounded if they are C -bounded for some constant C.
Source: Algebraic & Geometric Topology
Authors: Antoine Goldsborough, Stefanie Zbinden
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
OpenAlex ID: W4405954400