Treeable Equivalence Relations and Essential Countability

John D. Clemens

Treeable Equivalence Relations and Essential Countability

Southern Illinois University

Research output: Contribution to conference › Presentation

Abstract

<div class="line" id="line-5"> We establish a dichotomy theorem characterizing when a treeable Borel equivalence relation E is essentially countable. Under additional assumptions on the treeing, we show that E is essentially countable if and only if there is no continuous embedding of the relation &Eopf; 1 into E . In particular, we generalize and provide a classical proof of the analogous result for hypersmooth equivalence relations due to Kechris and Louveau. By considering a special class of treeings, we use our dichotomy to deduce several results about the global structure of the Borel reducibility hierarchy on equivalence relations, namely: the collection of treeable Borel equivalence relations is unbounded in the Borel-reducibility hierarchy; for every Borel equivalence relation which is not essentially hyperfinite we may find equivalence relations of arbitrarily high descriptive complexity with which it is incomparable under Borel reducibility; and for a sufficiently complicated Borel Wadge class Γ there is no minimum non-potentially Γ equivalence relation. This is joint work with Dominique Lecomte and Ben Miller</div>

Original language	American English
State	Published - 19 Apr 2015
Externally published	Yes
Event	Spring Western Sectional Meeting of the American Mathematical Society - Las Vegas, NV Duration: 19 Apr 2015 → …

Conference

Conference	Spring Western Sectional Meeting of the American Mathematical Society
Period	19/04/15 → …

EGS Disciplines

Mathematics

Cite this

@conference{c473ba620d1946a981822b8233a4979a,

title = "Treeable Equivalence Relations and Essential Countability",

abstract = " We establish a dichotomy theorem characterizing when a treeable Borel equivalence relation E is essentially countable. Under additional assumptions on the treeing, we show that E is essentially countable if and only if there is no continuous embedding of the relation &Eopf; 1 into E . In particular, we generalize and provide a classical proof of the analogous result for hypersmooth equivalence relations due to Kechris and Louveau. By considering a special class of treeings, we use our dichotomy to deduce several results about the global structure of the Borel reducibility hierarchy on equivalence relations, namely: the collection of treeable Borel equivalence relations is unbounded in the Borel-reducibility hierarchy; for every Borel equivalence relation which is not essentially hyperfinite we may find equivalence relations of arbitrarily high descriptive complexity with which it is incomparable under Borel reducibility; and for a sufficiently complicated Borel Wadge class Γ there is no minimum non-potentially Γ equivalence relation. This is joint work with Dominique Lecomte and Ben Miller",

author = "Clemens, {John D.}",

year = "2015",

month = apr,

day = "19",

language = "American English",

note = "Spring Western Sectional Meeting of the American Mathematical Society ; Conference date: 19-04-2015",

}

TY - CONF

T1 - Treeable Equivalence Relations and Essential Countability

AU - Clemens, John D.

PY - 2015/4/19

Y1 - 2015/4/19

N2 - We establish a dichotomy theorem characterizing when a treeable Borel equivalence relation E is essentially countable. Under additional assumptions on the treeing, we show that E is essentially countable if and only if there is no continuous embedding of the relation &Eopf; 1 into E . In particular, we generalize and provide a classical proof of the analogous result for hypersmooth equivalence relations due to Kechris and Louveau. By considering a special class of treeings, we use our dichotomy to deduce several results about the global structure of the Borel reducibility hierarchy on equivalence relations, namely: the collection of treeable Borel equivalence relations is unbounded in the Borel-reducibility hierarchy; for every Borel equivalence relation which is not essentially hyperfinite we may find equivalence relations of arbitrarily high descriptive complexity with which it is incomparable under Borel reducibility; and for a sufficiently complicated Borel Wadge class Γ there is no minimum non-potentially Γ equivalence relation. This is joint work with Dominique Lecomte and Ben Miller

AB - We establish a dichotomy theorem characterizing when a treeable Borel equivalence relation E is essentially countable. Under additional assumptions on the treeing, we show that E is essentially countable if and only if there is no continuous embedding of the relation &Eopf; 1 into E . In particular, we generalize and provide a classical proof of the analogous result for hypersmooth equivalence relations due to Kechris and Louveau. By considering a special class of treeings, we use our dichotomy to deduce several results about the global structure of the Borel reducibility hierarchy on equivalence relations, namely: the collection of treeable Borel equivalence relations is unbounded in the Borel-reducibility hierarchy; for every Borel equivalence relation which is not essentially hyperfinite we may find equivalence relations of arbitrarily high descriptive complexity with which it is incomparable under Borel reducibility; and for a sufficiently complicated Borel Wadge class Γ there is no minimum non-potentially Γ equivalence relation. This is joint work with Dominique Lecomte and Ben Miller

M3 - Presentation

T2 - Spring Western Sectional Meeting of the American Mathematical Society

Y2 - 19 April 2015

ER -

Treeable Equivalence Relations and Essential Countability

Abstract

Conference

EGS Disciplines

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Cite this