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Point processes, cost, and the growth of rank in locally compact groups
Abstract
Let G be a locally compact, second countable, unimodular group that is nondiscrete and noncompact. We explore the ergodic theory of invariant point processes on G. Our first result shows that every free probability measure preserving (pmp) action of G can be realized by an invariant point process. We then analyze the cost of pmp actions of G using this language. We show that among free pmp actions, the cost is maximal on the Poisson processes. This follows from showing that every free point process weakly factors onto any Poisson process and that the cost is monotone for weak factors, up to some restrictions. We apply this to show that G × ℤ has fixed price 1, solving a problem of Carderi. We also show that when G is a semisimple real Lie group, the rank gradient of any Farber sequence of lattices in G is dominated by the cost of the Poisson process of G. The same holds for the symmetric space X of G. This, in particular, implies that if the cost of the Poisson process of the hyperbolic 3-space ℍ3 vanishes, then the ratio of the Heegaard genus and the rank of a hyperbolic 3-manifold tends to infinity over arbitrary expander Farber sequences, in particular, the ratio can get arbitrarily large. On the other hand, if the cost of the Poisson process on ℍ3 does not vanish, it solves the cost versus L2 Betti problem of Gaboriau for countable equivalence relations.
... Sparse factor graphs of Poisson point processes with a unique infinite cluster. The notion of cost, and the fixed price problem, were extended to unimodular locally compact second countable groups via connected (equivariant) factor graphs of free invariant point processes on the group G byÁbert and Mellick [1]. For our purposes, it suffices to recall that Poisson point processes then have maximal cost among all free invariant point processes [1,Theorem 1.2]. ...
... The notion of cost, and the fixed price problem, were extended to unimodular locally compact second countable groups via connected (equivariant) factor graphs of free invariant point processes on the group G byÁbert and Mellick [1]. For our purposes, it suffices to recall that Poisson point processes then have maximal cost among all free invariant point processes [1,Theorem 1.2]. ...
... where Y is a Poisson point process of intensity 1 on G and Y 0 := Y ∪ {1 G } [1, Definition 4.1]. We may also assume that Y is equipped with iid Unif[0, 1] marks, see [1,Theorem 1.7 & 1.8]. ...
We show that the uniqueness thresholds for Poisson-Voronoi percolation in symmetric spaces of connected higher rank semisimple Lie groups with property (T) converge to zero in the low-intensity limit. This phenomenon is fundamentally different from situations in which Poisson-Voronoi percolation has previously been studied. Our approach builds on a recent breakthrough of Fraczyk, Mellick and Wilkens (arXiv:2307.01194) and provides an alternative proof strategy for Gaboriau's fixed price problem. As a further application of our result, we give a new class of examples of non-amenable Cayley graphs that admit factor of iid bond percolations with a unique infinite cluster and arbitrarily small expected degree, answering a question inspired by Hutchcroft-Pete (Invent. math. 221 (2020)).}
... The proof of Theorem 1.2 is essentially a blend of a dynamical variation of Gaboriau's weak normality argument with the techniques developed in [AM22] for handling cost of nondiscrete groups. The dynamical element is heavily inspired by the recent work [FMW23], and recovers some of its results. ...
... The definition of cost for lcscu groups first appears explicitly in [Car23], [AM22], and [Con+21], although it already appears all but in name in Proposition 4.3 of [KPV15], described as folklore. ...
... A fundamental motivation for considering cost for such groups is the following: if one can establish fixed price for an lcscu group G, then one proves fixed price for all lattices in G simultaneously. Further motivation is found by [Car23] and independently [AM22]: for groups of fixed price one, one also gets uniform vanishing of rank gradient for so-called Farber sequences of lattices. Recall that if Γ n < G is a sequence of lattices, its rank gradient is ...
Let be a semisimple real Lie group and another locally compact second countable unimodular group. We prove that has fixed price one if has higher rank, or if has rank one and is a p-adic split reductive group of rank at least one. As an application we resolve a question of Gaboriau showing has fixed price one. Inspired by the very recent work arXiv:2307.01194v1 [math.GT], we employ the method developed by the author and Mikl\'os Ab\'ert to show that all essentially free probability measure preserving actions of groups weakly factor onto the Cox process driven by their amenable subgroups. We then show that if an amenable subgroup can be found satisfying a double recurrence property then the Cox process driven by it has cost one.
... We now review the necessary background on the Poisson point process. For a self-contained exposition using the same notation, see [2]. For an introduction to the Poisson point process, see [16], and for a thorough treatment see [17] and [4]. ...
... is ergodic, and for a proof of the latter one can see [10] Theorem 3.8, item (2). The statement implies that G (M(Z), µ) is mixing, which implies ergodicity. ...
In this note, we resolve a question of D'Achille, Curien, Enriquez, Lyons, and \"Unel by showing that the cells of the ideal Poisson Voronoi tessellation are indistinguishable. This follows from an application of the Howe-Moore theorem and a theorem of Meyerovitch about the nonexistence of thinnings of the Poisson point process. We also give an alternative proof of Meyerovitch's theorem.
... To address this problem, with Sam Mellick [8], we recently introduced a cost theory for point processes of locally compact groups. Note that Alessandro Carderi has already introduced the cost of p.m.p. actions of locally compact groups in his nice paper [18] and used an ultraproduct language to prove that the maximal cost of a p.m.p. action of G dominates the rank gradient, at least for uniformly discrete Farber sequences of lattices. ...
... In the paper [8] we prove that the Poisson processes have maximal cost among free point processes and that this number dominates the rank gradient of any Farber sequence in G. This is an analogue of my theorem with Benjy Weiss for discrete groups [11], as Poisson processes are arguably the substitutes of iid actions in the locally compact setting. ...
... This result gives as a by-product the first examples of non trivial fixed price for connected lcsc groups (Definition A.9). In contrast, fixed price 1 for the direct product of some lcsc groups with the integers is obtained in [AM21]. Once a Haar measure is prescribed on G, the quantity cost(R B )−1 covolume(B) does not depend on the cross section B, since the restrictions are pairwise stably orbit equivalent (Proposition A.8). ...
... The cost of some lcsc groups is also considered in [AM21]. ...
We show that several new classes of groups are measure strongly treeable. In particular, finitely generated groups admitting planar Cayley graphs, elementarily free groups, and the group of isometries of the hyperbolic plane and all its closed subgroups. In higher dimensions, we also prove a dichotomy that the fundamental group of a closed aspherical 3-manifold is either amenable or has strong ergodic dimension 2. Our main technical tool is a method for finding measurable treeings of Borel planar graphs by constructing one-ended spanning subforests in their planar dual. Our techniques for constructing one-ended spanning subforests also give a complete classification of the locally finite pmp graphs which admit Borel a.e. one-ended spanning subforests.
... Also, this theorem can be generalized to the more general setting of actions of groups. This is closely related to the result of [1] showing that every essentially free probability-measure-preserving action of a non-discrete locally compact second-countable group G is isomorphic to a point process of finite intensity on G. ...
In this article, we show that every stationary random measure on R^d that is essentially free (i.e., has no symmetries a.s.) admits a point process as a factor. As a result, we improve the results of Last and Thorisson (2022) on the existence of a factor balancing allocation between ergodic pairs of stationary random measures Φ and Ψ with equal intensities. In particular, we prove that such an allocation exists if Φ is diffuse and either (Φ, Ψ) is essentially free or Φ assigns zero measure to every (d − 1)-dimensional affine hyperplane. The main result is deduced from an existing result in descriptive set theory, that is, the existence of lacunary sections. We also weaken the assumption of being essentially free to the case where a discrete group of symmetries is allowed.
... Some elements of G could belong to multiple Voronoi cells, although this cannot happen in R n . Abért and Mellick identify this distinction in [AM21] and introduce a tie-breaking function in pursuit of a true partition; we do the same. ...
A Poisson system is a Poisson point process and a group action, together forming a measure-preserving dynamical system. Ornstein and Weiss proved Poisson systems over many amenable groups were isomorphic in their 1987 paper. We consider Poisson systems over non-discrete, non-compact, locally compact Polish groups, and we prove by construction all Poisson systems over such a group are finitarily isomorphic, producing examples of isomorphisms for non-amenable group actions. As a corollary, we prove Poisson systems and products of Poisson systems are finitarily isomorphic.
We investigate the rank gradient and growth of torsion in homology in
residually finite groups. As a tool, we introduce a new complexity notion for
generating sets, using measured groupoids and combinatorial cost.
As an application we prove the vanishing of the above invariants for Farber
sequences of subgroups of right angled groups. A group is right angled if it
can be generated by a sequence of elements of infinite order such that any two
consecutive elements commute.
Most non-uniform lattices in higher rank simple Lie groups are right angled.
We provide the first examples of uniform (co-compact) right angled arithmetic
groups in and for some
values of p,q. This is a class of lattices for which the Congruence Subgroup
Property is not known in general.
Using rigidity theory and the notion of invariant random subgroups it follows
that both the rank gradient and the homology torsion growth vanish for an
arbitrary sequence of subgroups in any right angled lattice in a higher rank
simple Lie group.
The purpose of this note is to prove various versions of the ergodic decomposition theorem for probability measures on standard Borel spaces which are quasi-invariant under a Borel ac- tion of a locally compact second countable group or a discrete nonsingular equivalence relation. In the process we obtain a simul- taneous ergodic decomposition of all quasi-invariant probability measures with a prescribed Radon-Nikodym derivative, analogous to classical results about decomposition of invariant probability measures.
We study the asymptotic behavior of Betti numbers, twisted torsion and other
spectral invariants of sequences of locally symmetric spaces. Our main results
are uniform versions of the DeGeorge--Wallach Theorem, of a theorem of Delorme
and various other limit multiplicity theorems.
The idea is to adapt Benjamini-Schramm convergence (BS-convergence),
originally introduced for sequences of finite graphs of bounded degree, to
sequences of Riemannian manifolds. Exploiting the rigidity theory of higher
rank lattices, we show that when volume tends to infinity, higher rank locally
symmetric spaces BS-converge to their universal cover. We prove that
BS-convergence implies a convergence of certain spectral invariants, the
Plancherel measures. This leads to convergence of volume normalized
multiplicities of unitary representations and Betti numbers.
We also prove a strong quantitative version of BS-convergence for arbitrary
sequences of congruence covers of a fixed arithmetic manifold. This leads to
upper estimates on the rate of convergence of normalized Betti numbers in the
spirit of Sarnak-Xue.
An important role in our approach is played by the notion of Invariant Random
Subroups. For higher rank simple Lie groups G, exploiting rigidity theory,
and in particular the Nevo-Stuck-Zimmer theorem and Kazhdan`s property (T), we
are able to analyze the space of IRSs of G. In rank one, the space of IRSs is
much richer. We build some explicit 2 and 3-dimensional real hyperbolic IRSs
that are not induced from lattices and employ techniques of
Gromov--Piatetski-Shapiro to construct similar examples in higher dimension.
We show that for any countable group, any free probability measure preserving
action of the group weakly contains all Bernoulli actions of the group. It
follows that for a finitely generated groups, the cost is maximal on Bernoulli
actions and that all free factors of i.i.d.-s the group have the same cost.
We also show that if a probability measure preserving action f is ergodic,
but not strongly ergodic, then f is weakly equivalent to f\timesI where I
denotes the trivial action on the unit interval. This leads to a relative
version of the Glasner-Weiss dichotomy.
We show that the group of almost automorphisms of a d-regular tree does not
admit lattices. As far as we know this is the first such example among
(compactly generated) simple locally compact groups.
The title refers to the area of research which studies infinite groups using measure-theoretic tools, and studies the restrictions that group structure imposes on ergodic theory of their actions. The paper is a survey of recent developments focused on the notion of Measure Equivalence between groups, and Orbit Equivalence between group actions. We discuss known invariants and classification results (rigidity) in both areas. Comment: A survey for the Proceedings of a Conference honoring Robert Zimmer's 60th birthday. 61 pages. Referees comments are incorporated, in particular sections 3.1.2 (cost) and 3.1.4 (Cowling Haagerup invariant) were updated. Some new references added
We study the growth of the rank of subgroups of finite index in residually finite groups, by relating it to the notion of cost. As a by-product, we show that the `Rank vs. Heegaard genus' conjecture on hyperbolic 3-manifolds is incompatible with the `Fixed Price problem' in topological dynamics.
Let \Pi be an ergodic simple point process on R^d and let \Pi^* be its Palm version. Thorisson [Ann. Probab. 24 (1996) 2057-2064] proved that there exists a shift coupling of \Pi and \Pi^*; that is, one can select a (random) point Y of \Pi such that translating \Pi by -Y yields a configuration whose law is that of \Pi^*. We construct shift couplings in which Y and \Pi^* are functions of \Pi, and prove that there is no shift coupling in which \Pi is a function of \Pi^*. The key ingredient is a deterministic translation-invariant rule to allocate sets of equal volume (forming a partition of R^d) to the points of \Pi. The construction is based on the Gale-Shapley stable marriage algorithm [Amer. Math. Monthly 69 (1962) 9-15]. Next, let \Gamma be an ergodic random element of {0,1}^{Z^d} and let \Gamma^* be \Gamma conditioned on \Gamma(0)=1. A shift coupling X of \Gamma and \Gamma^* is called an extra head scheme. We show that there exists an extra head scheme which is a function of \Gamma if and only if the marginal E[\Gamma(0)] is the reciprocal of an integer. When the law of \Gamma is product measure and d\geq3, we prove that there exists an extra head scheme X satisfying E\exp c\|X\|^d<\infty; this answers a question of Holroyd and Liggett [Ann. Probab. 29 (2001) 1405-1425].
A factor graph of a point process is a graph whose vertices are the points of the process, and which is constructed from the process in a deterministic isometry-invariant way. We prove that the d-dimensional Poisson process has a one-ended tree as a factor graph. This implies that the Poisson points can be given an ordering isomorphic to the usual ordering of the integers in a deterministic isometry-invariant way. For d \geq 4 our result answers a question posed by Ferrari, Landim and Thorisson. We prove also that any isometry-invariant ergodic point process of finite intensity in Euclidean or hyperbolic space has a perfect matching as a factor graph provided all the inter-point distances are distinct.
Previous work showed that all Bernoulli shifts over a free group are orbit-equivalent. This result is strengthened here by replacing Bernoulli shifts with the wider class of properly ergodic countable state Markov chains over a free group. A list of related open problems is provided.
Starting around the late 1950s, several research communities began relating the geometry of graphs to stochastic processes on these graphs. This book, twenty years in the making, ties together research in the field, encompassing work on percolation, isoperimetric inequalities, eigenvalues, transition probabilities, and random walks. Written by two leading researchers, the text emphasizes intuition, while giving complete proofs and more than 850 exercises. Many recent developments, in which the authors have played a leading role, are discussed, including percolation on trees and Cayley graphs, uniform spanning forests, the mass-transport technique, and connections on random walks on graphs to embedding in Hilbert space. This state-of-the-art account of probability on networks will be indispensable for graduate students and researchers alike.
Let X be a higher rank symmetric space or a Bruhat-Tits building of dimension at least 2 such that the isometry group of X has property (T). We prove that for every torsion free lattice any homology class in has a representative cycle of total length . As an application we show that $\dim_{\mathbb F_2} H_1(\Gamma\backslash X,\mathbb F_2)=o_X({\rm Vol}(\Gamma\backslash X)).
This paper concerns the study of the global structure of measure-preserving actions of countable groups on standard probability spaces. Weak containment is a hierarchical notion of complexity of such actions, motivated by an analogous concept in the theory of unitary representations. This concept gives rise to an associated notion of equivalence of actions, called weak equivalence, which is much coarser than the notion of isomorphism (conjugacy). It is well understood now that, in general, isomorphism is a very complex notion, a fact which manifests itself, for example, in the lack of any reasonable structure in the space of actions modulo isomorphism. On the other hand, the space of weak equivalence classes is quite well behaved. Another interesting fact that relates to the study of weak containment is that many important parameters associated with actions, such as the type, cost, and combinatorial parameters, turn out to be invariants of weak equivalence and in fact exhibit desirable monotonicity properties with respect to the pre-order of weak containment, a fact that can be useful in certain applications. There has been quite a lot of activity in this area in the last few years, and our goal in this paper is to provide a survey of this work.
To every countable probability measure preserving (pmp) equivalence relation
and probability space one can canonically associate a
new countable pmp equivalence relation , called the Bernoulli
extension of with base space . We prove that if
is ergodic and non-amenable, and K is non-atomic, then contains the orbits of a free ergodic pmp action of the free group
. Moreover, we provide conditions which imply that this holds for
any non-trivial probability space K. The proof relies on an extension of
techniques of Gaboriau and Lyons \cite{GL07}.
As an application, we prove that every non-amenable ergodic pmp equivalence
relation admits uncountably many ergodic extensions which are pairwise not
(stably) von Neumann equivalent (hence, pairwise not orbit equivalent). We
further deduce that any non-amenable unimodular locally compact second
countable group admits uncountably many free ergodic pmp actions which are
pairwise not von Neumann equivalent.
The main result of the paper is classification of free multidimensional Borel
flows up to Lebesgue Orbit Equivalence, by which we understand an orbit
equivalence that preserves the Lebesgue measure on each orbit. Two non smooth
Euclidean flows are shown to be Lebesgue Orbit Equivalence if and only if they
admit the same number of invariant ergodic probability measures.
We show the rank (i.e. minimal size of a generating set) of lattices cannot grow faster than the volume.
Many phenomena in physics, chemistry, and biology can be modelled by spatial random processes. One such process is continuum percolation, which is used when the phenomenon being modelled is made up of individual events that overlap, for example, the way individual raindrops eventually make the ground evenly wet. This is a systematic rigorous account of continuum percolation. Two models, the Boolean model and the random connection model, are treated in detail, and related continuum models are discussed. All important techniques and methods are explained and applied to obtain results on the existence of phase transitions, equality and continuity of critical densities, compressions, rarefaction, and other aspects of continuum models. This self-contained treatment, assuming only familiarity with measure theory and basic probability theory, will appeal to students and researchers in probability and stochastic geometry.
We generalize two theorems about K-automorphisms from Z to all amenable groups with good entropy theory (this class includes all unimodular amenable groups which are not an increasing union of compact subgroups). The first theorem is that such actions are uniformly mixing; the second is that their spectrum is Lebesgue with countable multiplicity. For the proof we will develop an entropy theory for discrete amenable equivalence relations.
1. Introduction.- 2. Moore's Ergodicity Theorem.- 3. Algebraic Groups and Measure Theory.- 4. Amenability.- 5. Rigidity.- 6. Margulis' Arithmeticity Theorems.- 7. Kazhdan's Property (T).- 8. Normal Subgroups of Lattices.- 9. Further Results on Ergodic Actions.- 10. Generalizations to p-adic groups and S-arithmetic groups.- Appendices.- A. Borel spaces.- B. Almost everywhere identities on groups.- References.
We study in this paper graph coloring problems in the context of descriptive set theory. We consider graphs G=(X, R), where the vertex set X is a standard Borel space (i.e., a complete separable metrizable space equipped with its σ-algebra of Borel sets), and the edge relation R ⊆ X^2 is
"definable", i.e., Borel, analytic, co-analytic, etc.
A Borel n-coloring of such a graph, where 1⩽ n ⩽ N_0 , is a Borel map c: X → Y with card(Y)=n, such
that xRy⇒c(x) ≠ {c( y). If such a Borel coloring exists we define the Borel
chromatic number of G, in symbols X_B(G), to be the smallest such n.
Otherwise we say that G has uncountable Borel chromatic number, in symbols X_B(G) > N_0.
Thesis (Ph. D.)--Harvard University, 1973.
We examine three key conjectures in 3-manifold theory: the virtually Haken
conjecture, the positive virtual b_1 conjecture and the virtually fibred
conjecture. We explore the interaction of these conjectures with the following
seemingly unrelated areas: eigenvalues of the Laplacian, and Heegaard
splittings.
We first give a necessary and sufficient condition, in terms of spectral
geometry, for a finitely presented group to have a finite index subgroup with
infinite abelianisation. For negatively curved 3-manifolds, we show that this
is equivalent to a statement about generalised Heegaard splittings. We also
formulate a conjecture about the behaviour of Heegaard genus under finite
covers which, together with a conjecture of Lubotzky and Sarnak about Property
tau, would imply the virtually Haken conjecture for hyperbolic 3-manifolds.
Along the way, we prove a number of unexpected theorems about 3-manifolds. For
example, we show that for any closed 3-manifold that fibres over the circle
with pseudo-Anosov monodromy, any cyclic cover dual to the fibre of
sufficiently large degree has an irreducible, weakly reducible Heegaard
splitting. Also, we establish lower and upper bounds on the Heegaard genus of
the congruence covers of an arithmetic hyperbolic 3-manifold, which are linear
in volume.
We discuss an inequality for graphs, which relates the distances between components of any minimal cut set to the lengths of generators for the homology of the graph. Our motivation arises from percolation theory. In particular this result is applied to Cayley graphs of nite presentations of groups with one end, where it gives an exponential bound on the number of minimal cut sets, and thereby shows that the critical probability for percola- tion on these graphs is neither zero nor one. We further show for this same class of graphs that the critical probability for the coalescence of all innite components into a single one is neither zero nor one.
We prove that the L^2-Betti numbers of a unimodular locally compact group G
coincide, up to a natural scaling constant, with the L^2-Betti numbers of the
countable equivalence relation induced on a cross section of any essentially
free ergodic probability measure preserving action of G. As a consequence, we
obtain that the reduced and un-reduced L^2-Betti numbers of G agree and that
the L^2-Betti numbers of a lattice Gamma in G equal those of G up to scaling by
the covolume of Gamma in G. We also deduce several vanishing results, including
the vanishing of the reduced L^2-cohomology for amenable locally compact
groups.
It is known that for every second countable locally compact group G, there
exists a proper G-invariant metric which induces the topology of the group.
This is no longer true for coset spaces G/H viewed as G-spaces. We study
necessary and sufficient conditions which ensure the existence of such metrics
on G/H.
We study a new dynamical invariant for dicrete groups: the cost. It is a real number in {1−1/n}∪[1,∞], bounded by the number of generators of the group, and it is well behaved with respect to finite index subgroups.
Namely, the quantities 1 minus the cost are related by multiplying by the index. The cost of every infinite amenable group
equals 1. We compute it in some other situations, including free products, free products with amalgamation and HNN-extensions
over amenable groups and for direct product situations. For instance, the cost of the free group on n generators equals n. We prove that each possible finite value of the cost is achieved by a finitely generated group. It is dynamical because
it relies on measure preserving free actions on probability Borel spaces. In most cases, groups have fixed price, which implies that two freely acting groups which define the same orbit partition must have the same cost. It enables us
to distinguish the orbit partitions of probability-preserving free actions of free groups of different ranks. At the end of
the paper, we give a mercuriale, i.e. a list of costs of different groups. The cost is in fact an invariant of ergodic measure-preserving equivalence relations
and is defined using graphings. A treeing is a measurable way to provide every equivalence class (=orbit) with the structure of a simplicial tree, this an example
of graphing. Not every relation admits a treeing: we prove that every free action of a cost 1 non-amenable group is not treeable,
but we prove that subrelations of treeable relations are treeable. We give examples of relations which cannot be produced
by an action of any finitely generated group. The cost of a relation which can be decomposed as a direct product is shown
to be 1. We define the notion for a relation to be a free product or an HNN-extension and compute the cost for the resulting
relation from the costs of the building blocks. The cost is also an invariant of the pairs von Neumann algebra/Cartan subalgebra.
On etend certains des resultats fondamentaux de la theorie des isomorphismes aux actions des groupes unimodulaires moyennables generaux. Les groupes moyennables et leur propre entropie. Entropie des actions mesurables. Theoremes d'isomorphismes
We consider a Poisson process η on an arbitrary measurable space with an arbitrary sigma-finite intensity measure. We establish an explicit Fock space representation
of square integrable functions of η. As a consequence we identify explicitly, in terms of iterated difference operators, the integrands in the Wiener–Itô chaos
expansion. We apply these results to extend well-known variance inequalities for homogeneous Poisson processes on the line
to the general Poisson case. The Poincaré inequality is a special case. Further applications are covariance identities for
Poisson processes on (strictly) ordered spaces and Harris–FKG-inequalities for monotone functions of η.
KeywordsPoisson process–Chaos expansion–Derivative operator–Kabanov–Skorohod integral–Malliavin calculus–Poincaré inequality–Variance inequalities–Infinitely divisible random measure
Let G be a measurable group with Haar measure λ, acting properly on a space S and measurably on a space T. Then any σ-finite, jointly invariant measure M on S× T admits a disintegration nÄm{\nu \otimes \mu} into an invariant measure ν on S and an invariant kernel μ from S to T. Here we construct ν andμ by a general skew factorization, which extends an approach by Rother and Zähle for homogeneous spaces S over G. This leads to easy extensions of some classical propositions for invariant disintegration, previously known in the homogeneous
case. The results are applied to the Palm measures of jointly stationary pairs (ξ, η), where ξ is a random measure on S and η is a random element in T.
KeywordsInvariant disintegration–Orbit selection–Inversion kernel–Skew factorization–Duality–Stationary random measures–Palm kernel
We construct a counterexample to the Rank versus Genus Conjecture, i.e. a
closed orientable hyperbolic 3-manifold with rank of its fundamental group
smaller than its Heegaard genus. Moreover, we show that the discrepancy between
rank and Heegaard genus can be arbitrarily large for hyperbolic 3-manifolds. We
also construct toroidal such examples containing hyperbolic JSJ pieces.
We consider the Directed Spanning Forest (DSF) constructed as follows: given
a Poisson point process N on the plane, the ancestor of each point is the
nearest vertex of N having a strictly larger abscissa. We prove that the DSF is
actually a tree. Contrary to other directed forests of the literature, no
Markovian process can be introduced to study the paths in our DSF. Our proof is
based on a comparison argument between surface and perimeter from percolation
theory. We then show that this result still holds when the points of N
belonging to an auxiliary Boolean model are removed. Using these results, we
prove that there is no bi-infinite paths in the DSF.
Let × be a Poisson point process of intensity λ on the real line. A thickening of it is a (deterministic) measurable function f such that X∪f(X) is a Poisson point process of intensity λ′ where λ′ > λ. An equivariant thickening is a thickening which commutes with all shifts of the line. We show that a thickening exists but an equivariant thickening does not. We prove similar results for thickenings which commute only with integer shifts and in the discrete and multi-dimensional settings. This answers 3 questions of Holroyd, Lyons and Soo.
We briefly consider also a much more general setup in which we ask for the existence of a deterministic coupling satisfying a relation between two probability measures. We present a conjectured sufficient condition for the existence of such couplings.
We study isomorphism invariant point processes of whose groups of symmetries are almost surely trivial. We define a 1-ended, locally finite tree factor on the points of the process, that is, a mapping of the point configuration to a graph on it that is measurable and equivariant with the point process. This answers a question of Holroyd and Peres. The tree will be used to construct a factor isomorphic to . This perhaps surprising result (that any d and n works) solves a problem by Steve Evans. The construction, based on a connected clumping with vertices in each clump of the i'th partition, can be used to define various other factors. Comment: 9 pages
Given a homogeneous Poisson process on with intensity
, we prove that it is possible to partition the points into two sets,
as a deterministic function of the process, and in an isometry-equivariant way,
so that each set of points forms a homogeneous Poisson process, with any given
pair of intensities summing to . In particular, this answers a
question of Ball [Electron. Commun. Probab. 10 (2005) 60--69], who proved that
in d=1, the Poisson points may be similarly partitioned (via a
translation-equivariant function) so that one set forms a Poisson process of
lower intensity, and asked whether the same is possible for all d. We do not
know whether it is possible similarly to add points (again chosen as a
deterministic function of a Poisson process) to obtain a Poisson process of
higher intensity, but we prove that this is not possible under an additional
finitariness condition.
Incluye bibliografía e índice
We consider a lcsc group G acting properly on a Borel space S and measurably on an underlying sigma-finite measure space. Our first main result is a transport formula connecting the Palm pairs of jointly stationary random measures on S. A key (and new) technical result is a measurable disintegration of the Haar measure on G along the orbits. The second main result is an intrinsic characterization of the Palm pairs of a G-invariant random measure. We then proceed with deriving a general version of the mass-transport principle for possibly non-transitive and non-unimodular group operations first in a deterministic and then in its full probabilistic form. Comment: 26 pages
One-ended spanning subforests and treeability of groups
- C T Conley
- D Gaboriau
- A S Marks
- R D Tucker-Drob
Lecture Notes on Random Geometric Models-Random Graphs, Point Processes and Stochastic Geometry
- B Blaszczyszyn
The palm groupoid of a point process and factor graphs on amenable and property (T) groups
- S Mellick
Around the orbit equivalence theory of the free groups, cost and ℓ2 Betti numbers
- D Gaboriau