51 research outputs found
Krieger's finite generator theorem for actions of countable groups I
For an ergodic probability-measure-preserving action of a countable group , we define the Rokhlin entropy
to be the infimum of the Shannon entropies of
countable generating partitions. It is known that for free ergodic actions of
amenable groups this notion coincides with classical Kolmogorov--Sinai entropy.
It is thus natural to view Rokhlin entropy as a close analogue to classical
entropy. Under this analogy we prove that Krieger's finite generator theorem
holds for all countably infinite groups. Specifically, if
then there exists a generating partition
consisting of sets
Every action of a non-amenable group is the factor of a small action
It is well known that if is a countable amenable group and factors onto , then the
entropy of the first action must be greater than or equal to the entropy of the
second action. In particular, if has infinite
entropy, then the action does not admit any
finite generating partition. On the other hand, we prove that if is a
countable non-amenable group then there exists a finite integer with the
following property: for every probability-measure-preserving action there is a -invariant probability measure
on such that factors onto . For many non-amenable groups, can be chosen to
be or smaller. We also obtain a similar result with respect to continuous
actions on compact spaces and continuous factor maps
Weak containment and Rokhlin entropy
We define a new notion of weak containment for joinings, and we show that
this notion implies an inequality between relative Rokhlin entropies. This
leads to new upper bounds to Rokhlin entropy. We also use this notion to study
how Pinsker algebras behave under direct products, and we study the Rokhlin
entropy of restricted actions of finite-index subgroups.Comment: References update
Burnside's Problem, spanning trees, and tilings
In this paper we study geometric versions of Burnside's Problem and the von
Neumann Conjecture. This is done by considering the notion of a
translation-like action. Translation-like actions were introduced by Kevin
Whyte as a geometric analogue of subgroup containment. Whyte proved a geometric
version of the von Neumann Conjecture by showing that a finitely generated
group is non-amenable if and only if it admits a translation-like action by any
(equivalently every) non-abelian free group. We strengthen Whyte's result by
proving that this translation-like action can be chosen to be transitive when
the acting free group is finitely generated. We furthermore prove that the
geometric version of Burnside's Problem holds true. That is, every finitely
generated infinite group admits a translation-like action by . This answers
a question posed by Whyte. In pursuit of these results we discover an
interesting property of Cayley graphs: every finitely generated infinite group
has some Cayley graph having a regular spanning tree. This regular spanning
tree can be chosen to have degree 2 (and hence be a bi-infinite Hamiltonian
path) if and only if has finitely many ends, and it can be chosen to have
any degree greater than 2 if and only if is non-amenable. We use this last
result to then study tilings of groups. We define a general notion of
polytilings and extend the notion of MT groups and ccc groups to the setting of
polytilings. We prove that every countable group is poly-MT and every finitely
generated group is poly-ccc.Comment: 24 pages; 1 figure; minor revision
A subgroup formula for f-invariant entropy
We study a measure entropy for finitely generated free group actions called
f-invariant entropy. The f-invariant entropy was developed by Lewis Bowen and
is essentially a special case of his measure entropy theory for actions of
sofic groups. In this paper we relate the f-invariant entropy of a finitely
generated free group action to the f-invariant entropy of the restricted action
of a subgroup. We show that the ratio of these entropies equals the index of
the subgroup. This generalizes a well known formula for the Kolmogorov--Sinai
entropy of amenable group actions. We then extend the definition of f-invariant
entropy to actions of finitely generated virtually free groups. We also obtain
a numerical virtual measure conjugacy invariant for actions of finitely
generated virtually free groups.Comment: Corollary 1.3 has been removed due to an error in its proof.
Corollary 1.2 is ne
Cost, -Betti numbers and the sofic entropy of some algebraic actions
In 1987, Ornstein and Weiss discovered that the Bernoulli -shift over the
rank two free group factors onto the seemingly larger Bernoulli -shift. With
the recent creation of an entropy theory for actions of sofic groups (in
particular free groups), their example shows the surprising fact that entropy
can increase under factor maps. In order to better understand this phenomenon,
we study a natural generalization of the Ornstein--Weiss map for countable
groups. We relate the increase in entropy to the cost and to the first
-Betti number of the group. More generally, we study coboundary maps
arising from simplicial actions and, under certain assumptions, relate
-Betti numbers to the failure of the Juzvinski{\u\i} addition formula.
This work is built upon a study of entropy theory for algebraic actions. We
prove that for actions on profinite groups via continuous group automorphisms,
topological sofic entropy is equal to measure sofic entropy with respect to
Haar measure whenever the homoclinic subgroup is dense. For algebraic actions
of residually finite groups we find sufficient conditions for the sofic entropy
to be equal to the supremum exponential growth rate of periodic points
Krieger's finite generator theorem for actions of countable groups III
We continue the study of Rokhlin entropy, an isomorphism invariant for
probability-measure-preserving actions of countable groups introduced in Part
I. In this paper we prove a non-ergodic finite generator theorem and use it to
establish sub-additivity and semi-continuity properties of Rokhlin entropy. We
also obtain formulas for Rokhlin entropy in terms of ergodic decompositions and
inverse limits. Finally, we clarify the relationship between Rokhlin entropy,
sofic entropy, and classical Kolmogorov--Sinai entropy. In particular, using
Rokhlin entropy we give a new proof of the fact that ergodic actions with
positive sofic entropy have finite stabilizers.Comment: Minor revision
Locally Nilpotent Groups and Hyperfinite Equivalence Relations
A long standing open problem in the theory of hyperfinite equivalence
relations asks if the orbit equivalence relation generated by a Borel action of
a countable amenable group is hyperfinite. In this paper we show that this
question has a positive answer when the acting group is locally nilpotent. This
extends previous results obtained by Gao-Jackson for abelian groups and by
Jackson-Kechris-Louveau for finitely generated nilpotent-by-finite groups. Our
proof is based on a mixture of coarse geometric properties of locally nilpotent
groups together with an adaptation of the Gao-Jackson machinery
Arbitrarily Large Residual Finiteness Growth
The residual finiteness growth of a group quantifies how well approximated
the group is by its finite quotients. In this paper, we construct groups with
arbitrarily large residual finiteness growth. We also demonstrate a new
relationship between residual finiteness growth and some decision problems in
groups, which we apply to our new groups.Comment: 6 page
The Koopman representation and positive Rokhlin entropy
In a prior paper, the author generalized the classical factor theorem of
Sinai to actions of arbitrary countably infinite groups. In the present paper,
we use this theorem and the techniques of its proof in order to study
connections between positive entropy phenomena and the Koopman representation.
Following the line of work initiated by Hayes for sofic entropy, we show in a
certain precise manner that all positive entropy must come from portions of the
Koopman representation that embed into the left-regular representation. By
combining this result with the generalized factor theorem of the previous
paper, we conclude that for actions having completely positive outer entropy,
the Koopman representation must be isomorphic to the countable direct sum of
the left-regular representation. This generalizes a theorem of Dooley--Golodets
for countable amenable groups. As a final consequence, we observe that actions
with completely positive outer entropy must be mixing, and when the group is
non-amenable they must be strongly ergodic and have spectral gap. We also prove
generalizations of these results that apply relative to sub--algebras
and apply to non-free actions
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