Skip to main content Accessibility help

Local and doubly empirical convergence and the entropy of algebraic actions of sofic groups

  • BEN HAYES (a1)


Let $G$ be a sofic group and $X$ a compact group with $G\curvearrowright X$ by automorphisms. Using (and reformulating) the notion of local and doubly empirical convergence developed by Austin, we show that in many cases the topological and the measure-theoretic entropy with respect to the Haar measure of $G\curvearrowright X$ agree. Our method of proof recovers all known examples. Moreover, the proofs are direct and do not go through explicitly computing the measure-theoretic or topological entropy.



Hide All
[1] Austin, T.. Additivity properties of sofic entropy and measures on model spaces. Forum Math. Sigma 4(e25) (2016), 79 pp.
[2] Austin, T.. An asymptotic equipartition property for measures on model spaces. Preprint, 2017,arXiv:1701.08723, Trans. Amer. Math. Soc., to appear.
[3] Berg, K.. Convolution of invariant measures, maximal entropy. Math. Systems Theory 3 (1969), 146150.
[4] Bowen, L.. Measure conjugacy invariants for actions of countable sofic groups. J. Amer. Math. Soc. 23 (2010), 217245.
[5] Bowen, L.. Entropy for expansive algebraic actions of residually finite groups. Ergod. Th. & Dynam. Sys. 31(3) (2011), 703718.
[6] Bowen, L.. Entropy theory for sofic groupoids I: the foundations. J. Anal. Math. 124(1) (2014), 149233.
[7] Bowen, L. and Li, H.. Harmonic models and spanning forests of residually finite groups. J. Funct. Anal. 263(7) (2012), 17691808.
[8] Deninger, C.. Fuglede–Kadison determinants and entropy for actions of discrete amenable groups. J. Amer. Math. Soc. 19 (2006), 737758.
[9] Gaboriau, D. and Seward, B.. Cost, $\ell ^{2}$ -Betti numbers and the sofic entropy of some algebraic actions. Preprint, 2015, arXiv:1509.02482, J. Anal. Math., to appear.
[10] Hayes, B.. Independence tuples and Deninger’s problem. Groups Geom. Dyn., to appear.
[11] Hayes, B.. Mixing and spectral gap relative to Pinsker factors for sofic groups. Proceedings in Honor of Vaughan F. R. Jones 60th Birthday Conferences, to appear.
[12] Hayes, B.. Polish models and sofic entropy. J. Inst. Math. Jussieu, to appear.
[13] Hayes, B.. Fuglede–Kadison determinants and sofic entropy. Geom. Funct. Anal. 26(2) (2016), 520606.
[14] Kerr, D.. Sofic measure entropy via finite partitions. Groups Geom. Dyn. 7 (2013), 617632.
[15] Kerr, D. and Li, H.. Topological entropy and the variational principle for actions of sofic groups. Invent. Math. 186 (2011), 501558.
[16] Kerr, D. and Li, H.. Soficity, amenability, and dynamical entropy. Amer. J. Math. 135(3) (2013), 721761.
[17] Kowalski, E.. An Introduction to the Representation Theory of Groups (Graduate Studies in Mathematics, 155) . American Mathematical Society, Providence, RI, 2014.
[18] Li, H.. Sofic mean dimension. Adv. Math. 244 (2014), 570604.
[19] Li, H. and Liang, B.. Sofic mean length. Preprint, 2015, arXiv:1510.07655.
[20] Li, H., Peterson, J. and Schmidt, K.. Ergodicity of principal algebraic group actions. Recent Trends in Ergodic Theory and Dynamical Systems (Contemporary Mathematics, 631) . American Mathematical Society, Providence, RI, 2015, pp. 201210.
[21] Lück, W.. L 2 -Invariants: Theory and Applications to Geometry and K-theory. Springer, Berlin, 2002.
[22] Schmidt, K.. Dynamical Systems of Algebraic Origin (Progress in Mathematics, 128) . Birkhäuser, Basel, 1995.

Related content

Powered by UNSILO

Local and doubly empirical convergence and the entropy of algebraic actions of sofic groups

  • BEN HAYES (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.