Essential Countability of Treeable Equivalence Relations

John D. Clemens; Dominique Lecomte; Benjamin D. Miller

doi:10.1016/j.aim.2014.07.035

Essential Countability of Treeable Equivalence Relations

John D. Clemens
, Dominique Lecomte
, Benjamin D. Miller

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

<div class="line" id="line-5"> We establish a dichotomy theorem characterizing the circumstances under which a treeable Borel equivalence relation <i> E </i> is essentially countable. Under additional topological assumptions on the treeing, we in fact show that E is essentially countable if and only if there is no continuous embedding of <span style='color: rgb(51, 51, 51); background-color: rgb(249, 249, 249); font-family: "Open Sans", "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px;'> &Eopf; </span> 1 into <i> E </i> . Our techniques also yield the first classical proof of the analogous result for hypersmooth equivalence relations, and allow us to show that up to continuous Kakutani embeddability, there is a minimum Borel function which is not essentially countable-to-one.</div>

Original language	American English
Journal	Advances in Mathematics
Volume	265
DOIs	https://doi.org/10.1016/j.aim.2014.07.035
State	Published - 10 Nov 2014
Externally published	Yes

Keywords

dichotomy theorem
essentially countable

EGS Disciplines

Mathematics

Access to Document

10.1016/j.aim.2014.07.035License: Unspecified

Cite this

@article{2c7f7ab3d4664847aba8aeb066776567,

title = "Essential Countability of Treeable Equivalence Relations",

abstract = " We establish a dichotomy theorem characterizing the circumstances under which a treeable Borel equivalence relation E is essentially countable. Under additional topological assumptions on the treeing, we in fact show that E is essentially countable if and only if there is no continuous embedding of \&Eopf; 1 into E . Our techniques also yield the first classical proof of the analogous result for hypersmooth equivalence relations, and allow us to show that up to continuous Kakutani embeddability, there is a minimum Borel function which is not essentially countable-to-one.",

keywords = "dichotomy theorem, essentially countable",

author = "Clemens, \{John D.\} and Dominique Lecomte and Miller, \{Benjamin D.\}",

year = "2014",

month = nov,

day = "10",

doi = "10.1016/j.aim.2014.07.035",

language = "American English",

volume = "265",

}

TY - JOUR

T1 - Essential Countability of Treeable Equivalence Relations

AU - Clemens, John D.

AU - Lecomte, Dominique

AU - Miller, Benjamin D.

PY - 2014/11/10

Y1 - 2014/11/10

N2 - We establish a dichotomy theorem characterizing the circumstances under which a treeable Borel equivalence relation E is essentially countable. Under additional topological assumptions on the treeing, we in fact show that E is essentially countable if and only if there is no continuous embedding of &Eopf; 1 into E . Our techniques also yield the first classical proof of the analogous result for hypersmooth equivalence relations, and allow us to show that up to continuous Kakutani embeddability, there is a minimum Borel function which is not essentially countable-to-one.

AB - We establish a dichotomy theorem characterizing the circumstances under which a treeable Borel equivalence relation E is essentially countable. Under additional topological assumptions on the treeing, we in fact show that E is essentially countable if and only if there is no continuous embedding of &Eopf; 1 into E . Our techniques also yield the first classical proof of the analogous result for hypersmooth equivalence relations, and allow us to show that up to continuous Kakutani embeddability, there is a minimum Borel function which is not essentially countable-to-one.

KW - dichotomy theorem

KW - essentially countable

UR - https://doi.org/10.1016/j.aim.2014.07.035

U2 - 10.1016/j.aim.2014.07.035

DO - 10.1016/j.aim.2014.07.035

M3 - Article

VL - 265

JO - Advances in Mathematics

JF - Advances in Mathematics

ER -

Essential Countability of Treeable Equivalence Relations

Abstract

Keywords

EGS Disciplines

Access to Document

Other files and links

Fingerprint

Cite this