Тема: On scaling entropy for group actions

In this paper, we study generic p.m.p. actions of amenable groups. The main object we focus on is the scaling entropy of an action — the invariant of slow entropy type proposed by A. Vershik. This invariant is based on the dynamics of measurable metrics on the underlying measure space and reflects the asymptotic behavior of the minimal epsilon- net of the averaged metric. The scaling entropy invariant was studied. We will give all the necessary definitions in Section 2.2. More details are given in a recent survey.
It turns out that some properties of the scaling entropy of a generic action can be established. In particular, we show that its asymptotic behavior can not be bounded from above by any nontrivial bound. For the case of a single transformation, similar results were obtained. Typical behavior of other slow entropy type invariants in the context of generic actions and generic extensions was studied. Our result gives, together with the results from, the negative answer to Weiss’ question about the existence of a universal zero entropy system for all amenable groups.
Also, we study lower bounds for the generic growth rate of scaling entropy. In the case of a residually finite group, a similar result holds true: there exists no non-constant lower bound for the scaling entropy of a generic action. However, it is not true in general. It turns out that there exist discrete amenable groups that have a scaling entropy growth gap meaning that the scaling entropy of any free p.m.p. action of such a group has to grow faster than some fixed unbounded function. We show an example of such a group in Section 5.2. Our example bases on the theory of growth in finite groups, in particular the growth theorem by H. Helfgott and its generalizations from.
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In view of Section 5.1 and Theorem 3, one may wonder if it is always the case that the scaling entropy of a generic action grows arbitrarily slow (along a subsequence, of course). We already know that it is true provided the group possesses a compact free action, but it is unclear for groups without such actions. We say that a group G has a scaling entropy growth gap with respect to equipment A if there exists a function ф(п) tending to infinity such that H(a, A) £ ф for every free p.m.p. action a of the group G. In this section we show that there exists a group with a scaling entropy growth gap.
Let G = SL(2, Fp) be the group of all 2 x 2-matrices with determinant 1 over the algebraic closure of a finite field Fp, where p > 2 is a prime number. Clearly, G is countable, and it can be presented as a union of increasing finite subgroups G U? 1 Gn, where each Gn = SL(2, Fqn) and Fqn is a finite extension of Fqn_1.
We will use the following Growth Theorem, initially proved in [6] by H. Helfgott for SL(2, Fp) and then generalized to the following result (see [17]).
Theorem 13. Let L be a finite simple group of Lie type of rank r and A a generating set of L. Then either A3 = L or |A3| > c| A 1''’, where c and 5 depend only on r.
Theorem 14. The group G = SL(2, Fp) with equipment A = {Gn} admits scaling entropy growth gap. The function ф(п) = log(qn) is the desired lower bound.
Proof. Consider a free p.m.p. action G ^ (X,p). Take some non-trivial element g0 from G1 = SL(2,Fp), let us take g0 = (J0, for instance. Since g0 has order p and the action is free, there exists a measurable partition f of (X, p) into p cells such that f(x) = £((до)гх) for every i = 1,... ,p — 1. That is, each cell of f contains exactly one point from each g0-orbit. Let p^ be the cut semimetric corresponding to f.
Suppose that He2 (X,p,Gfvp--) < log k and let X0, X1,..., Xk be the corresponding decomposition. Since Gn is finite, the measure space decomposes as (Gn, v) x (Y, n), where the action of Gn preserves the second component. Since the exceptional set X0 has measure less than e2, the n-measure of those y-s that satisfy |Gn x {y} П X01 > ' Gn | is less than e. The restriction of G2vp£ to each Gn-orbit is Gn-invariant and can be obtained by averaging the restriction of p^. The restriction of p^ to a Gn-orbit corresponds to its partition into p parts of equal size. Hence, the restriction of p-. has mean value at least 2 as well as its average since averaging preserves L1-norm. All of the above implies that there exits at least one Gn-orbit with an invariant metric that has e-entropy (with respect to uniform measure) less than log k and L1-norm of at least 1. It suffices to prove the following.
Claim 15. Let p be a left-invariant semimetric on SL(2, Fq) with diameter greater than 3e, where e 2 (1, 2). Then H£(SL(2, Fq),v,p) ^ clogq, where v is the uniform probability measure and c is an absolute constant.
Indeed, we can identify the orbit that we found above with the group SL(2, Fqn) with the left-invariant semimetric which has diameter at least 2. Applying Claim 15, we obtain log k ^ c log qn and complete the proof.
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