2019, 15: 133-141. doi: 10.3934/jmd.2019016

The work of Lewis Bowen on the entropy theory of non-amenable group actions

Laboratoire de Probabilités, Statistique et Modélisation, Sorbonne Université, 4 Place Jussieu, 75252 Paris Cedex 05, France

Received  April 05, 2019

We present the achievements of Lewis Bowen, or, more precisely, his breakthrough works after which a theory started to develop. The focus will therefore be made here on the isomorphism problem for Bernoulli actions of countable non-amenable groups which he solved brilliantly in two remarkable papers. Here two invariants were introduced, which led to many developments.

Citation: Jean-Paul Thouvenot. The work of Lewis Bowen on the entropy theory of non-amenable group actions. Journal of Modern Dynamics, 2019, 15: 133-141. doi: 10.3934/jmd.2019016
References:
[1]

A. Alpeev, On Pinsker factors for Rokhlin entropy, J. Math. Sci. (N.Y.), 209 (2015), 826-829.  doi: 10.1007/s10958-015-2529-8.  Google Scholar

[2]

T. Austin and P. Burton, Uniform mixing and completely positive sofic entropy, to appear in J. Anal. Math. Google Scholar

[3]

T. Austin, The geometry of model spaces for probability-preserving actions of sofic groups, Anal. Geom. Metr. Spaces, 4 (2014), 160-186.  doi: 10.1515/agms-2016-0006.  Google Scholar

[4]

T. Austin, Additivity properties of sofic entropy and measures on model spaces, Forum Math. Sigma, 4 (2016), e25, 79 pp. doi: 10.1017/fms.2016.18.  Google Scholar

[5]

L. Bowen, A measure-conjugacy invariant for free group actions, Ann. of Math. (2), 171 (2010), 1387-1400.  doi: 10.4007/annals.2010.171.1387.  Google Scholar

[6]

L. Bowen, Measure conjugacy invariants for actions of countable sofic groups, J. Amer. Math. Soc., 23 (2010), 217-245.  doi: 10.1090/S0894-0347-09-00637-7.  Google Scholar

[7]

L. Bowen, The ergodic theory of free group actions: Entropy and the f-invariant, Groups Geom. Dyn., 4 (2010), 419-432.  doi: 10.4171/GGD/89.  Google Scholar

[8]

L. Bowen, Weak isomorphisms of Bernoulli shifts, Israel J. Math., 183 (2011), 93-102.  doi: 10.1007/s11856-011-0043-3.  Google Scholar

[9]

L. Bowen, Every countably infinite group is almost Ornstein, in Dynamical Systems and Group Actions, Contemp. Math., 567, Amer. Math. Soc., Providence, RI, 2012, 67–78. doi: 10.1090/conm/567/11234.  Google Scholar

[10]

L. Bowen, Sofic entropy and amenable groups, Ergodic Theory Dynam. Systems, 32 (2012), 427-466.  doi: 10.1017/S0143385711000253.  Google Scholar

[11]

L. Bowen, Finitary random interlacements and the Gaboriau-Lyons problem, preprint, arXiv: 1707.09573v3. Google Scholar

[12]

L. Bowen, Sofic homological invariants and the weak Pinsker property, arXiv: 1807.08191. Google Scholar

[13]

E. Gordon and A. Vershik, Groups that are locally embeddable in the class of finite groups, Algebra i Analiz, 9 (1997), 71-97.   Google Scholar

[14]

M. Gromov, Endomorphisms of symbolic algebraic varieties, J. Eur. Math. Soc. (JEMS), 1 (1999), 109-197.  doi: 10.1007/PL00011162.  Google Scholar

[15]

B. Hayes, Fuglede-Kadison determinants and sofic entropy, Geom. Funct. Anal., 26 (2016), 520-606.  doi: 10.1007/s00039-016-0370-y.  Google Scholar

[16]

B. Hayes, Mixing and spectral gap relative to Pinsker factor for sofic groups, in Proceedings of the 2014 Maui and 2015 Qinhuangdao Conferences in Honour of Vaughan F. R. Jones' 60th Birthday, Proc. Centre Math. Appl. Austral. Nat. Univ., 46, Austral. Nat. Univ., Canberra, 2017,193–221.  Google Scholar

[17]

B. Hayes, Sofic entropy of Gaussian actions, Ergodic Theory Dynam. Systems, 37 (2017), 2187-2222.  doi: 10.1017/etds.2016.6.  Google Scholar

[18]

D. Kerr, Sofic measure entropy via finite partitions, Groups Geom. Dyn., 7 (2013), 617-632.  doi: 10.4171/GGD/200.  Google Scholar

[19]

D. Kerr, Bernoulli actions of sofic groups have completey positive entropy, Israel J. Math., 202 (2014), 461-474.  doi: 10.1007/s11856-014-1077-0.  Google Scholar

[20]

D. Kerr and H. Li, Bernoulli actions and infinite entropy, Groups Geom. Dyn., 5 (2011), 663-672.  doi: 10.4171/GGD/142.  Google Scholar

[21]

D. Kerr and H. Li, Entropy and the variational principle for actions of sofic groups, Invent. Math., 186 (2011), 501-558.  doi: 10.1007/s00222-011-0324-9.  Google Scholar

[22]

D. Kerr and H. Li, Soficity, amenability, and dynamical entropy, Amer. J. Math., 135 (2013), 721-761.  doi: 10.1353/ajm.2013.0024.  Google Scholar

[23]

J. Kieffer, A generalized Shannon-McMillan theorem for the action of an amenable group on a probability space, Ann. Probability, 3 (1975), 1031-1037.  doi: 10.1214/aop/1176996230.  Google Scholar

[24]

D. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Analyse Math., 48 (1987), 1-141.  doi: 10.1007/BF02790325.  Google Scholar

[25]

D. Ornstein and B. Weiss, Ergodic theory of amenable group actions. I. The Rohlin lemma, Bull. Amer. Math. Soc. (N.S.), 2 (1980), 161-164.  doi: 10.1090/S0273-0979-1980-14702-3.  Google Scholar

[26]

S. Popa, Some computations of 1-cohomology groups and construction of non-orbit-equivalent actions, J. Inst. Math. Jussieu, 5 (2006), 309-332.  doi: 10.1017/S1474748006000016.  Google Scholar

[27]

B. Seward, Bernoulli shifts with base of equal entropy are isomorphic, arXiv: 1805.08279, 2018. Google Scholar

[28]

B. Seward, Krieger's finite generator theorem for ergodic actions of countable groups I, Invent. Math., 215 (2019), 265-310.  doi: 10.1007/s00222-018-0826-9.  Google Scholar

[29]

B. Seward, The Koopman representation and positive Rokhlin entropy, arXiv: 1804.05270, 2018. Google Scholar

[30]

B. Seward, Positive entropy actions of countable groups factor onto Bernoulli shifts, to appear in J. Amer. Math. Soc. Google Scholar

[31]

A. Stepin, Bernoulli shifts on groups, Dokl. Akad. Nauk SSSR, 223 (1975), 300-302.   Google Scholar

[32]

B. Weiss, Entropy and actions of sofic groups, Discrete Contin. Dynam. Syst. Ser. B, 20 (2015), 3375-3383.  doi: 10.3934/dcdsb.2015.20.3375.  Google Scholar

show all references

References:
[1]

A. Alpeev, On Pinsker factors for Rokhlin entropy, J. Math. Sci. (N.Y.), 209 (2015), 826-829.  doi: 10.1007/s10958-015-2529-8.  Google Scholar

[2]

T. Austin and P. Burton, Uniform mixing and completely positive sofic entropy, to appear in J. Anal. Math. Google Scholar

[3]

T. Austin, The geometry of model spaces for probability-preserving actions of sofic groups, Anal. Geom. Metr. Spaces, 4 (2014), 160-186.  doi: 10.1515/agms-2016-0006.  Google Scholar

[4]

T. Austin, Additivity properties of sofic entropy and measures on model spaces, Forum Math. Sigma, 4 (2016), e25, 79 pp. doi: 10.1017/fms.2016.18.  Google Scholar

[5]

L. Bowen, A measure-conjugacy invariant for free group actions, Ann. of Math. (2), 171 (2010), 1387-1400.  doi: 10.4007/annals.2010.171.1387.  Google Scholar

[6]

L. Bowen, Measure conjugacy invariants for actions of countable sofic groups, J. Amer. Math. Soc., 23 (2010), 217-245.  doi: 10.1090/S0894-0347-09-00637-7.  Google Scholar

[7]

L. Bowen, The ergodic theory of free group actions: Entropy and the f-invariant, Groups Geom. Dyn., 4 (2010), 419-432.  doi: 10.4171/GGD/89.  Google Scholar

[8]

L. Bowen, Weak isomorphisms of Bernoulli shifts, Israel J. Math., 183 (2011), 93-102.  doi: 10.1007/s11856-011-0043-3.  Google Scholar

[9]

L. Bowen, Every countably infinite group is almost Ornstein, in Dynamical Systems and Group Actions, Contemp. Math., 567, Amer. Math. Soc., Providence, RI, 2012, 67–78. doi: 10.1090/conm/567/11234.  Google Scholar

[10]

L. Bowen, Sofic entropy and amenable groups, Ergodic Theory Dynam. Systems, 32 (2012), 427-466.  doi: 10.1017/S0143385711000253.  Google Scholar

[11]

L. Bowen, Finitary random interlacements and the Gaboriau-Lyons problem, preprint, arXiv: 1707.09573v3. Google Scholar

[12]

L. Bowen, Sofic homological invariants and the weak Pinsker property, arXiv: 1807.08191. Google Scholar

[13]

E. Gordon and A. Vershik, Groups that are locally embeddable in the class of finite groups, Algebra i Analiz, 9 (1997), 71-97.   Google Scholar

[14]

M. Gromov, Endomorphisms of symbolic algebraic varieties, J. Eur. Math. Soc. (JEMS), 1 (1999), 109-197.  doi: 10.1007/PL00011162.  Google Scholar

[15]

B. Hayes, Fuglede-Kadison determinants and sofic entropy, Geom. Funct. Anal., 26 (2016), 520-606.  doi: 10.1007/s00039-016-0370-y.  Google Scholar

[16]

B. Hayes, Mixing and spectral gap relative to Pinsker factor for sofic groups, in Proceedings of the 2014 Maui and 2015 Qinhuangdao Conferences in Honour of Vaughan F. R. Jones' 60th Birthday, Proc. Centre Math. Appl. Austral. Nat. Univ., 46, Austral. Nat. Univ., Canberra, 2017,193–221.  Google Scholar

[17]

B. Hayes, Sofic entropy of Gaussian actions, Ergodic Theory Dynam. Systems, 37 (2017), 2187-2222.  doi: 10.1017/etds.2016.6.  Google Scholar

[18]

D. Kerr, Sofic measure entropy via finite partitions, Groups Geom. Dyn., 7 (2013), 617-632.  doi: 10.4171/GGD/200.  Google Scholar

[19]

D. Kerr, Bernoulli actions of sofic groups have completey positive entropy, Israel J. Math., 202 (2014), 461-474.  doi: 10.1007/s11856-014-1077-0.  Google Scholar

[20]

D. Kerr and H. Li, Bernoulli actions and infinite entropy, Groups Geom. Dyn., 5 (2011), 663-672.  doi: 10.4171/GGD/142.  Google Scholar

[21]

D. Kerr and H. Li, Entropy and the variational principle for actions of sofic groups, Invent. Math., 186 (2011), 501-558.  doi: 10.1007/s00222-011-0324-9.  Google Scholar

[22]

D. Kerr and H. Li, Soficity, amenability, and dynamical entropy, Amer. J. Math., 135 (2013), 721-761.  doi: 10.1353/ajm.2013.0024.  Google Scholar

[23]

J. Kieffer, A generalized Shannon-McMillan theorem for the action of an amenable group on a probability space, Ann. Probability, 3 (1975), 1031-1037.  doi: 10.1214/aop/1176996230.  Google Scholar

[24]

D. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Analyse Math., 48 (1987), 1-141.  doi: 10.1007/BF02790325.  Google Scholar

[25]

D. Ornstein and B. Weiss, Ergodic theory of amenable group actions. I. The Rohlin lemma, Bull. Amer. Math. Soc. (N.S.), 2 (1980), 161-164.  doi: 10.1090/S0273-0979-1980-14702-3.  Google Scholar

[26]

S. Popa, Some computations of 1-cohomology groups and construction of non-orbit-equivalent actions, J. Inst. Math. Jussieu, 5 (2006), 309-332.  doi: 10.1017/S1474748006000016.  Google Scholar

[27]

B. Seward, Bernoulli shifts with base of equal entropy are isomorphic, arXiv: 1805.08279, 2018. Google Scholar

[28]

B. Seward, Krieger's finite generator theorem for ergodic actions of countable groups I, Invent. Math., 215 (2019), 265-310.  doi: 10.1007/s00222-018-0826-9.  Google Scholar

[29]

B. Seward, The Koopman representation and positive Rokhlin entropy, arXiv: 1804.05270, 2018. Google Scholar

[30]

B. Seward, Positive entropy actions of countable groups factor onto Bernoulli shifts, to appear in J. Amer. Math. Soc. Google Scholar

[31]

A. Stepin, Bernoulli shifts on groups, Dokl. Akad. Nauk SSSR, 223 (1975), 300-302.   Google Scholar

[32]

B. Weiss, Entropy and actions of sofic groups, Discrete Contin. Dynam. Syst. Ser. B, 20 (2015), 3375-3383.  doi: 10.3934/dcdsb.2015.20.3375.  Google Scholar

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