The expected degree of minimal spanning forests

Thom, Andreas

doi:10.1007/s00493-014-3160-x

Cited by 6 publications

(9 citation statements)

References 17 publications

(35 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We note that [GM15] is representative of a lot of recent interest in measure-theoretic versions of von Neumann's problem spawned by the pioneering work [GL07]. Thus, the main result of [GL07] was strengthened in [Ku13], and its proof was simplified in [Th13]. Very recently, von Neumann's problem was shown to have a positive solution for non-amenable equivalence relations that act on hyperbolic bundles [Bo15].…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

von Neumann’s problem and extensions of non-amenable equivalence relations

2018

View full text Add to dashboard Cite

The goals of this paper are twofold. First, we generalize the result of Gaboriau and Lyons [GL07] to the setting of von Neumann's problem for equivalence relations, proving that for any non-amenable ergodic probability measure preserving (pmp) equivalence relation R, the Bernoulli extension over a non-atomic base space (K, κ) contains the orbit equivalence relation of a free ergodic pmp action of F2. Moreover, we provide conditions which imply that this holds for any non-trivial probability space K. Second, we use this result to prove that any non-amenable unimodular locally compact second countable group admits uncountably many free ergodic pmp actions which are pairwise not von Neumann equivalent (hence, pairwise not orbit equivalent).F 2 by [GL07, Theorem 1], it follows that the same is true for R K . However, Theorem A is new whenever R does not arise as the orbit equivalence relation of a free pmp action of a countable group (see [Fu99] for examples of such R). Also, note that if R = R(Γ [0, 1] Γ ), then R K is isomorphic to R. Theorem A implies that R contains the orbits of a free ergodic pmp action of F 2 , for any non-amenable Γ, and therefore recovers [GL07, Theorem 1].Remark 1.2. At the end of [GL07], the authors posed the following analogue of von Neumann's problem for equivalence relations: does every ergodic non-amenable countable pmp equivalence relation R contain R(F 2 X) for some free ergodic pmp action of F 2 ? The main result of [GL07] shows that this is indeed the case if R arises from the Bernoulli action with base ([0, 1], λ) of a non-amenable countable group. Theorem A shows that, more generally, this holds for the Bernoulli extension with base ([0, 1], λ) of any ergodic non-amenable countable pmp equivalence relation.We turn now to the second main result of this paper and to the history motivating it. In the early 1980s, D. Ornstein and B. Weiss [OW80], extending work of H. Dye [Dy59], showed that any two ergodic pmp actions of countable infinite amenable groups are orbit equivalent. Moreover, as a consequence of [CFW81], all free properly ergodic pmp actions of a unimodular amenable lcsc group G are pairwise orbit equivalent. On the other hand, over the next two decades, several families of non-amenable countable groups, including property (T) groups [Hj02] and non-abelian free groups [GP03], were shown to admit uncountably many actions which are pairwise not orbit equivalent.Unifying many of these results, it was shown in [Io06] that any countable group Γ containing a copy of F 2 has uncountably many free ergodic actions which are pairwise not orbit equivalent. Thus nearly three decades after the solution to von Neumann's problem [Ol80], the relationship between general non-amenable groups and the prototypical example of F 2 came again into focus. Gaboriau and Lyons' result in [GL07] was followed shortly by [Ep07], in which I. Epstein combined [GL07] with the methods of [Io06] via a new co-induction construction for group actions, proving that any countable non-amenable group Γ admits unc...

show abstract

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

von Neumann’s problem and extensions of non-amenable equivalence relations

2018

View full text Add to dashboard Cite

show abstract

“…By Proposition 5.3(h), to prove Theorem 5.8, it suffices to show that there exists a finite subset S ′ of G containing an element of infinite order g such that the unoriented Cayley graph Γ ′ = Cay uo (G, S ′ ∪ (S ′ ) −1 ) admits a G-invariant random spanning forest which has expected degree > 4 and almost surely contains all edges of Γ ′ labeled by g ±1 . The construction of such a forest given below is almost identical to the construction from the proof of [33,Lemma 5].…”

Section: 32mentioning

confidence: 98%

“…There are several standard constructions of G-invariant random spanning forests on Cayley graphs including the free minimal spanning forest (we refer the reader to [33] for the definition which will not be important to us).…”

mentioning

confidence: 99%

See 1 more Smart Citation

The Tarski numbers of groups

Ershov

Golan

Sapir

2015

Advances in Mathematics

View full text Add to dashboard Cite

The Tarski number of a non-amenable group G is the minimal number of pieces in a paradoxical decomposition of G. In this paper we investigate how Tarski numbers may change under various group-theoretic operations. Using these estimates and known properties of Golod-Shafarevich groups, we show that the Tarski numbers of 2-generated non-amenable groups can be arbitrarily large. We also use the cost of group actions to show that there exist groups with Tarski numbers 5 and 6. These provide the first examples of non-amenable groups without free subgroups whose Tarski number has been computed precisely. (M. Ershov), gili.golan@math.biu.ac.il (G. Golan), m.sapir@vanderbilt.edu (M. Sapir).

show abstract

“…The measure MSF(G, S) is called the minimal spanning forest of Cay(G, S). A theorem of Thom states that if G is non-amenable then the expected degree of MSF(G, S) is unbounded as the generating set S varies [33,34].…”

Section: Upper Bounds To Rokhlin Entropymentioning

confidence: 99%