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Entropy and isoperimetry for linear and non-linear group actions. (English) Zbl 1280.20043

Summary of results: 1. We shall prove in 2.2, generalizing Elek’s \(l_2\)-theorem, that
the action of every non-amenable group \(\Gamma\) on the space \(l_p(\Gamma)\) of \(p\)-summable real functions is \(I_*\)-linear for all \(1\leq p<\infty\).
We define in 2.3 a class of groups \(\Gamma\), which includes the free groups \(F_k\) for \(k\geq 2\) and many (but not all) non-amenable groups that contain no \(F_2\), such that their group algebras are \(\mathcal F\)-non-amenable (or \(I_{\mathcal F}\)-linear) in the following sense.
There exists a finite family of subgroups \(\Gamma_i\subset\Gamma\) in each group \(\Gamma\) from this class such that every linear subspace \(D\) in the group algebra of \(\Gamma\) over every field \(\mathbb F\) has \[ \max_i|\mathcal F\partial_{\Gamma_i}(D)|\geq\varepsilon|D| \] for some \(\varepsilon=\varepsilon(\Gamma)>0\).
2. We show in 3.2 (elaborating upon a remark by Dima Grigorev who pointed out to me how the polynomial Brunn-Minkowski inequality reduces to the combinatorial one with an order on the set of monomials) that
if \(\Gamma\) admits a left invariant order, then the linear algebraic profile equals the combinatorial one, \[ I_*(r;\Gamma)=I_\circ(r;\Gamma), \] and prove some related results.
3. We combine in 6.2 the Coulon-Saloff-Coste argument with the \(l_2\)-Loomis-Whitney inequality and prove, for example, that
the Følner function \(F_*(n)\) of the Hilbert space \(l_2(\Gamma)\) of square summable functions on every polycyclic non-virtually nilpotent group \(\Gamma\) has exponential growth.
(For the spaces of functions with finite supports this follows directly from the Coulhon-Saloff-Coste inequality and 2.)
4. We shall apply the entropic formalism (which mimics the martingale method in isoperimetry) to the group algebras of normal extensions and thus prove, for example, that
Grigorchuk groups (which may have subexponential growth \(G_\circ\prec\exp(n^\alpha)\) for \(0<\alpha<1\) and may be pure torsion) satisfy \[ F_*(n)\succ\exp(n^\beta)\text{ for some }\beta>0 \] (see 7.1, 7.2). Also we prove in 7.2 that
the (twice or more) iterated wreath products \(\Gamma\) of infinite groups, e.g. of (pure torsion amenable) Aleshin-Grigorchuk groups, have \[ F_*(n)\succ\exp(\exp(n)). \] 5. We shall exhibit in 8.1, 8.2 a class of amenable groups \(\Gamma\), where
the combinatorial Følner functions \(F_\circ(n;\Gamma)\) may grow arbitrarily fast
(these \(\Gamma\) are extensions of locally finite groups as in the last remark in Section 3 of A. Erschler [Geom. Dedicata 100, 157-171 (2003; Zbl 1049.20024)]) and, yet,
all these \(\Gamma\) have bounded linear algebraic profiles, \[ I_*(r;\Gamma)\leq\text{const}(\Gamma). \] On the other hand we produce
orderable amenable groups \(\Gamma\) where both combinatorial and linear algebraic Følner functions grow arbitrarily fast.
(These \(\Gamma\) are extensions of locally nilpotent groups rather than of locally finite groups that were suggested in [loc. cit.]. Possibly, one can also render orderable the construction of groups with subexponential growth from [A. Erschler, Duke Math. J. 134, No. 3, 591-613 (2006; Zbl 1159.20019)] and make not only \(F_\circ\), as it is done in [loc. cit.], but also \(F_*\) grow arbitrarily fast.)
6. We present in 9.1-9.3 a translation of the combinatorial argument from [A. Erschler, Geom. Dedicata, op. cit.] to the language of partitions, which improves the lower bounds from [loc. cit.] on Følner functions of the iterated wreath products and related classes of groups.
7. We shall prove in 10.1-10.3 some Brunn-Minkovski type inequalities for discrete groups and their group algebras.
Our presentation is self-contained and a significant part of the paper is expository.

MSC:

20F65 Geometric group theory
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
20F05 Generators, relations, and presentations of groups
20E22 Extensions, wreath products, and other compositions of groups
43A07 Means on groups, semigroups, etc.; amenable groups
51F99 Metric geometry
52A37 Other problems of combinatorial convexity
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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References:

[1] F. Almgren, Optimal isoperimetric inequalities. Indiana Univ. Math. J. 35 (1986), 451-547. · Zbl 0585.49030 · doi:10.1512/iumj.1986.35.35028
[2] L. Bartholdi, On amenability of group algebras, I. Israel J. Math. 168 (2008), 153-165. · Zbl 1167.43001 · doi:10.1007/s11856-008-1061-7
[3] A. Belov and R. Mikhailov, Free algebras of lie algebras close to nilpotent. Preprint 2008. · Zbl 1260.17007 · doi:10.4171/GGD/73
[4] T. Coulhon and L. Saloff-Coste, Isopérimétrie pour les groupes et les variétés. Rev. Mat. Iberoamericana 9 (1993), 293-314. · Zbl 0782.53066 · doi:10.4171/RMI/138
[5] E. G. Effros, New perspectives and some celebrated quantum inequalities. Preprint 2008.
[6] G. Elek, The amenability of affine algebras. J. Algebra 264 (2003), 469-478. · Zbl 1022.43001 · doi:10.1016/S0021-8693(03)00163-7
[7] G. Elek, On the analytic zero divisor conjecture of Linnell. Bull. London Math. Soc. 35 (2003), 236-238. · Zbl 1027.20002 · doi:10.1112/S002460930200173X
[8] G. Elek, The amenability and non-amenability of skew fields. Proc. Amer. Math. Soc. 134 (2006), 637-644. · Zbl 1152.12002 · doi:10.1090/S0002-9939-05-08128-1
[9] A. Erschler, On isoperimetric profiles of finitely generated groups. Geom. Dedicata 100 (2003), 157-171. · Zbl 1049.20024 · doi:10.1023/A:1025849602376
[10] A. Erschler, Piecewise automatic groups. Duke Math. J. 134 (2006), 591-613. · Zbl 1159.20019 · doi:10.1215/S0012-7094-06-13435-X
[11] E. Friedgut, Hypergraphs, entropy and inequalities. Amer. Math. Monthly 111 (2004), 749-760. · Zbl 1187.94017 · doi:10.2307/4145187
[12] R. J. Gardner, The Brunn-Minkowski inequality. Bull. Amer. Math. Soc. ( N.S.) 39 (2002), 355-405. · Zbl 1019.26008 · doi:10.1090/S0273-0979-02-00941-2
[13] M. Gromov, Convex sets and Kähler manifolds. In Advances in differential geometry and topology , World Scientific, Singapore 1990, 1-38. · Zbl 0770.53042
[14] M. Gromov, Topological invariants of dynamical systems and spaces of holomorphic maps: I. Math. Phys. Anal. Geom. 2 (1999), 323-415. · Zbl 1160.37322 · doi:10.1023/A:1009841100168
[15] M. Gromov, Random walk in random groups. Geom. Funct. Anal. 13 (2003), 73-146. · Zbl 1122.20021 · doi:10.1007/s000390300002
[16] M. Gromov, Singularities, expanders, hyperbolic geometry and fiberwise homology of maps. In preparation. 593
[17] M. Gromov, Mendelian dynamics and Sturtevant’s paradigm. In Geometric and proba- bilistic structures in dynamics , Contemp. Math. Ser., Amer. Math. Soc., Providence. RI, to appear. · Zbl 1207.92036
[18] M. Gromov and V. Milman, Generalization of the spherical isoperimetric inequality to uniformly convex Banach spaces. Compositio Math. 62 (1987), 263-282. · Zbl 0623.46007
[19] S. Hoory, N. Linial, and A. Wigderson, Expander graphs and their applications. Bull. Amer. Math. Soc. ( N.S.) 43 (2006), 439-561. · Zbl 1147.68608 · doi:10.1090/S0273-0979-06-01126-8
[20] O. E. Lanford III, Entropy and equilibrium states in classical statistical mechanics. In Statistical mechanics and mathematical problems (Battelle Seattle 1971 Rencontres), Lecture Notes in Phys. 20, Springer-Verlag, Berlin 1973, 1-113.
[21] P. W. Michor, D. Petz, and A. Andai, On the curvature of a certain Riemannian space of matrices. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3 (2000), 199-212. · Zbl 1031.53109 · doi:10.1142/S0219025700000145
[22] D. Witte Morris, Amenable groups that act on the line. Algebr. Geom. Topol. 6 (2006), 2509-2518. · Zbl 1185.20042 · doi:10.2140/agt.2006.6.2509
[23] M. Ohya and D. Petz, Quantum entropy and its use . Corr. 2nd printing, Texts and Mono- graphs in Physics, Springer-Verlag, Berlin 2004. · Zbl 0891.94008
[24] D. Petz and C. Sudár, Extending the Fisher metric to density matrices. In Geometry of present days science (Aarhus, Denmark, 1997), World Scientific, Singapore 1999, 21-33. · Zbl 0939.62008
[25] D. Rolfsen, Ordered groups and topology. Lecture Notes, Luminy, June 2001. · Zbl 1362.20001
[26] M. B. Ruskai, Another short and elementary proof of strong subadditivity of quantum entropy. Rep. Math. Phys. 60 (2007), 1-12. · Zbl 1140.82009 · doi:10.1016/S0034-4877(07)00019-5
[27] C. Pittet, and L. Saloff-Coste, Amenable groups, isoperimetric profiles and random walks. In Geometric group theory down under (Canberra, 1996), Walter de Gruyter, Berlin 1999, 293-316. · Zbl 0934.43001
[28] N. Th. Varopoulos, L. Saloff-Coste, and T. Coulhon, Analysis and geometry on groups . Cambridge Tracts in Math. 100, Cambridge University Press, Cambridge 1992. · Zbl 0813.22003
[29] A. Vershik, Amenability and approximation of infinite groups. Selecta Math. Soviet. 2 (1982), 311-330. · Zbl 0533.22007
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