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The Geometry of Model Spaces for Probability-Preserving Actions of Sofic Groups

100%

Austin T.

Analysis and Geometry in Metric Spaces

2016

tom 4

nr 1

Bowen’s notion of sofic entropy is a powerful invariant for classifying probability-preserving actions of sofic groups. It can be defined in terms of the covering numbers of certain metric spaces associated to such an action, the ‘model spaces’. The metric geometry of these model spaces can exhibit various interesting features, some of which provide other invariants of the action. This paper explores an approximate connectedness property of the model spaces, and uses it give a new proof that certain groups admit factors of Bernoulli shifts which are not Bernoulli. This was originally proved by Popa. Our proof covers fewer examples than his, but provides additional information about this phenomenon.

The Quest for A Characterization of Hom-Properties of Finite Character

58%

Broere I., Matsoha M. D. V., Heidema J.

Discussiones Mathematicae Graph Theory

2016

tom 36

nr 2

479-500

A graph property is a set of (countable) graphs. A homomorphism from a graph G to a graph H is an edge-preserving map from the vertex set of G into the vertex set of H; if such a map exists, we write G → H. Given any graph H, the hom-property →H is the set of H-colourable graphs, i.e., the set of all graphs G satisfying G → H. A graph property P is of finite character if, whenever we have that F ∈ P for every finite induced subgraph F of a graph G, then we have that G ∈ P too. We explore some of the relationships of the property attribute of being of finite character to other property attributes such as being finitely-induced-hereditary, being finitely determined, and being axiomatizable. We study the hom-properties of finite character, and prove some necessary and some sufficient conditions on H for →H to be of finite character. A notable (but known) sufficient condition is that H is a finite graph, and our new model-theoretic proof of this compactness result extends from hom-properties to all axiomatizable properties. In our quest to find an intrinsic characterization of those H for which →H is of finite character, we find an example of an infinite connected graph with no finite core and chromatic number 3 but with hom-property not of finite character.

Ograniczanie wyników

1 Analysis and Geometry in Metric Spaces

1 Discussiones Mathematicae Graph Theory

1 Austin T.

1 Broere I.

1 Heidema J.

1 Matsoha M. D. V.

2 2016

Wyniki wyszukiwania

The Geometry of Model Spaces for Probability-Preserving Actions of Sofic Groups

The Quest for A Characterization of Hom-Properties of Finite Character