Amenability and geometry of semigroups

Gray, Robert; Kambites, Mark

doi:10.1090/tran/6939

Amenability and geometry of semigroups
HTML articles powered by AMS MathViewer

by Robert D. Gray and Mark Kambites PDF

Trans. Amer. Math. Soc. 369 (2017), 8087-8103 Request permission

Abstract:

We study the connection between amenability, Følner conditions and the geometry of finitely generated semigroups. Using results of Klawe, we show that within an extremely broad class of semigroups (encompassing all groups, left cancellative semigroups, finite semigroups, compact topological semigroups, inverse semigroups, regular semigroups, commutative semigroups and semigroups with a left, right or two-sided zero element), left amenability coincides with the strong Følner condition. Within the same class, we show that a finitely generated semigroup of subexponential growth is left amenable if and only if it is left reversible. We show that the (weak) Følner condition is a left quasi-isometry invariant of finitely generated semigroups, and hence that left amenability is a left quasi-isometry invariant of left cancellative semigroups. We also give a new characterisation of the strong Følner condition in terms of the existence of weak Følner sets satisfying a local injectivity condition on the relevant translation action of the semigroup.

References

L. N. Argabright and C. O. Wilde, Semigroups satisfying a strong Følner condition, Proc. Amer. Math. Soc. 18 (1967), 587–591. MR 210797, DOI 10.1090/S0002-9939-1967-0210797-2
Vitaly Bergelson and Neil Hindman, Quotient sets and density recurrent sets, Trans. Amer. Math. Soc. 364 (2012), no. 9, 4495–4531. MR 2922599, DOI 10.1090/S0002-9947-2012-05417-1
Tullio Ceccherini-Silberstein and Michel Coornaert, Cellular automata and groups, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010. MR 2683112, DOI 10.1007/978-3-642-14034-1
Tullio Ceccherini-Silberstein and Michel Coornaert, The Myhill property for cellular automata on amenable semigroups, Proc. Amer. Math. Soc. 143 (2015), no. 1, 327–339. MR 3272758, DOI 10.1090/S0002-9939-2014-12227-1
Ferran Cedó and Jan Okniński, Semigroups of matrices of intermediate growth, Adv. Math. 212 (2007), no. 2, 669–691. MR 2329316, DOI 10.1016/j.aim.2006.11.004
Fan Chung, Laplacians and the Cheeger inequality for directed graphs, Ann. Comb. 9 (2005), no. 1, 1–19. MR 2135772, DOI 10.1007/s00026-005-0237-z
Joel M. Cohen, Cogrowth and amenability of discrete groups, J. Funct. Anal. 48 (1982), no. 3, 301–309. MR 678175, DOI 10.1016/0022-1236(82)90090-8
H. G. Dales, A. T.-M. Lau, and D. Strauss, Banach algebras on semigroups and on their compactifications, Mem. Amer. Math. Soc. 205 (2010), no. 966, vi+165. MR 2650729, DOI 10.1090/S0065-9266-10-00595-8
Mahlon M. Day, Amenable semigroups, Illinois J. Math. 1 (1957), 509–544. MR 92128
John Donnelly, Necessary and sufficient conditions for a subsemigroup of a cancellative left amenable semigroup to be left amenable, Int. J. Algebra 7 (2013), no. 5-8, 237–243. MR 3056449, DOI 10.12988/ija.2013.13023
Étienne Ghys and Pierre de la Harpe, Infinite groups as geometric objects (after Gromov), Ergodic theory, symbolic dynamics, and hyperbolic spaces (Trieste, 1989) Oxford Sci. Publ., Oxford Univ. Press, New York, 1991, pp. 299–314. MR 1130180
Robert Gray and Mark Kambites, A Švarc-Milnor lemma for monoids acting by isometric embeddings, Internat. J. Algebra Comput. 21 (2011), no. 7, 1135–1147. MR 2863430, DOI 10.1142/S021819671100687X
Robert Gray and Mark Kambites, Groups acting on semimetric spaces and quasi-isometries of monoids, Trans. Amer. Math. Soc. 365 (2013), no. 2, 555–578. MR 2995365, DOI 10.1090/S0002-9947-2012-05868-5
Robert D. Gray and Mark Kambites, Quasi-isometry and finite presentations of left cancellative monoids, Internat. J. Algebra Comput. 23 (2013), no. 5, 1099–1114. MR 3096313, DOI 10.1142/S0218196713500185
Robert D. Gray and Mark Kambites, A strong geometric hyperbolicity property for directed graphs and monoids, J. Algebra 420 (2014), 373–401. MR 3261466, DOI 10.1016/j.jalgebra.2014.08.007
R. I. Grigorchuk, Symmetrical random walks on discrete groups, Multicomponent random systems, Adv. Probab. Related Topics, vol. 6, Dekker, New York, 1980, pp. 285–325. MR 599539
R. I. Grigorchuk, Semigroups with cancellations of degree growth, Mat. Zametki 43 (1988), no. 3, 305–319, 428 (Russian); English transl., Math. Notes 43 (1988), no. 3-4, 175–183. MR 941053, DOI 10.1007/BF01138837
Harry Kesten, Full Banach mean values on countable groups, Math. Scand. 7 (1959), 146–156. MR 112053, DOI 10.7146/math.scand.a-10568
Maria Klawe, Semidirect product of semigroups in relation to amenability, cancellation properties, and strong Følner conditions, Pacific J. Math. 73 (1977), no. 1, 91–106. MR 470609
Alan L. T. Paterson, Amenability, Mathematical Surveys and Monographs, vol. 29, American Mathematical Society, Providence, RI, 1988. MR 961261, DOI 10.1090/surv/029
L. M. Shneerson, Polynomial growth in semigroup varieties, J. Algebra 320 (2008), no. 6, 2218–2279. MR 2437499, DOI 10.1016/j.jalgebra.2008.03.028
J. R. Sorenson, Left-amenable semigroups and cancellation, Notices of Amer. Math. Soc. 11 (1964), 763.
John Raymond Sorenson, EXISTENCE OF MEASURES THAT ARE INVARIANT UNDER A SEMIGROUP OF TRANSFORMATIONS, ProQuest LLC, Ann Arbor, MI, 1966. Thesis (Ph.D.)–Purdue University. MR 2616074
Yuji Takahashi, A counterexample to Sorenson’s conjecture: the finitely generated case, Semigroup Forum 67 (2003), no. 3, 429–431. MR 2002945, DOI 10.1007/s00233-001-0170-y
J. von Neumann, Zur allgemeinen theorie des maßes, Fund. Math. 13 (1929), 73–111.
Benjamin Willson, Reiter nets for semidirect products of amenable groups and semigroups, Proc. Amer. Math. Soc. 137 (2009), no. 11, 3823–3832. MR 2529892, DOI 10.1090/S0002-9939-09-09957-2
Zhuocheng Yang, Følner numbers and Følner type conditions for amenable semigroups, Illinois J. Math. 31 (1987), no. 3, 496–517. MR 892183