A new measure of growth for groups and algebras

Hołubowski, Waldemar

doi:10.1090/S1061-0022-08-01009-1

A new measure of growth for groups and algebras
HTML articles powered by AMS MathViewer

by Waldemar Hołubowski
Translated by: N. A. Vavilov

St. Petersburg Math. J. 19 (2008), 545-560

DOI: https://doi.org/10.1090/S1061-0022-08-01009-1

Published electronically: May 9, 2008

PDF | Request permission

Abstract:

The notion of a bandwidth growth is introduced, which generalizes the growth of groups and the bandwidth dimension, first discussed by J. Hannah and K. C. O’Meara for countable-dimensional algebras. The new measure of growth is based on certain infinite matrix representations and on the notion of growth of nondecreasing functions on the set of natural numbers. Two natural operations are defined on the set $\Omega ^{\star }$ of growths. With respect to these operations, $\Omega ^{\star }$ forms a lattice with many interesting algebraic properties; for example, $\Omega ^{\star }$ is distributive and dense and has uncountable antichains.

This new notion of growth is applied in order to define bandwidth growth for subgroups and subalgebras of infinite matrices and to study its properties.

References

Gene Abrams and Juan Jacobo Simón, Isomorphisms between infinite matrix rings: a survey, Algebra and its applications (Athens, OH, 1999) Contemp. Math., vol. 259, Amer. Math. Soc., Providence, RI, 2000, pp. 1–12. MR 1778492, DOI 10.1090/conm/259/04085
E. Arbarello, C. De Concini, and V. G. Kac, The infinite wedge representation and the reciprocity law for algebraic curves, Theta functions—Bowdoin 1987, Part 1 (Brunswick, ME, 1987) Proc. Sympos. Pure Math., vol. 49, Amer. Math. Soc., Providence, RI, 1989, pp. 171–190. MR 1013132, DOI 10.1090/pspum/049.1/1013132
Frank W. Anderson and Kent R. Fuller, Rings and categories of modules, 2nd ed., Graduate Texts in Mathematics, vol. 13, Springer-Verlag, New York, 1992. MR 1245487, DOI 10.1007/978-1-4612-4418-9
Stanley Burris and H. P. Sankappanavar, A course in universal algebra, Graduate Texts in Mathematics, vol. 78, Springer-Verlag, New York-Berlin, 1981. MR 648287
Victor Camillo, F. J. Costa-Cano, and J. J. Simón, Relating properties of a ring and its ring of row and column finite matrices, J. Algebra 244 (2001), no. 2, 435–449. MR 1859035, DOI 10.1006/jabr.2001.8901
Richard G. Cooke, Infinite Matrices and Sequence Spaces, Macmillan & Co., Ltd., London, 1950. MR 0040451
Roy O. Davies, Michael P. Drazin, and Mark L. Roberts, Universal properties of infinite matrices, J. Algebra 180 (1996), no. 2, 402–411. MR 1378536, DOI 10.1006/jabr.1996.0073
John D. Dixon, Most finitely generated permutation groups are free, Bull. London Math. Soc. 22 (1990), no. 3, 222–226. MR 1041134, DOI 10.1112/blms/22.3.222
É. Ghys and P. de la Harpe (eds.), Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Mathematics, vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1990 (French). Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988. MR 1086648, DOI 10.1007/978-1-4684-9167-8
K. R. Goodearl, P. Menal, and J. Moncasi, Free and residually Artinian regular rings, J. Algebra 156 (1993), no. 2, 407–432. MR 1216477, DOI 10.1006/jabr.1993.1082
R. I. Grigorchuk, On the Milnor problem of group growth, Dokl. Akad. Nauk SSSR 271 (1983), no. 1, 30–33 (Russian). MR 712546
R. I. Grigorchuk, Degrees of growth of finitely generated groups and the theory of invariant means, Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), no. 5, 939–985 (Russian). MR 764305
Mikhael Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53–73. MR 623534
C. K. Gupta and W. Hołubowski, Automorphisms of a free group of infinite rank, Algebra i Analiz 19 (2007), no. 2, 74–85; English transl., St. Petersburg Math. J. 19 (2008), no. 2, 215–223. MR 2333897, DOI 10.1090/S1061-0022-08-00994-1
—, Automorphisms of rooted tree of countable valency (in preparation).
Alexander J. Hahn and O. Timothy O’Meara, The classical groups and $K$-theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 291, Springer-Verlag, Berlin, 1989. With a foreword by J. Dieudonné. MR 1007302, DOI 10.1007/978-3-662-13152-7
John Hannah and K. C. O’Meara, A new measure of growth for countable-dimensional algebras, Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 2, 223–227. MR 1215312, DOI 10.1090/S0273-0979-1993-00427-0
John Hannah and K. C. O’Meara, A new measure of growth for countable-dimensional algebras. I, Trans. Amer. Math. Soc. 347 (1995), no. 1, 111–136. MR 1282887, DOI 10.1090/S0002-9947-1995-1282887-9
Waldemar Hołubowski, Free subgroups of the group of infinite unitriangular matrices, Internat. J. Algebra Comput. 13 (2003), no. 1, 81–86. MR 1970869, DOI 10.1142/S0218196703001341
Waldemar Hołubowski, Most finitely generated subgroups of infinite unitriangular matrices are free, Bull. Austral. Math. Soc. 66 (2002), no. 3, 419–423. MR 1939204, DOI 10.1017/S0004972700040272
Waldemar Hołubowski, The ubiquity of free subsemigroups of infinite triangular matrices, Semigroup Forum 66 (2003), no. 2, 231–235. MR 1953501, DOI 10.1007/s002330010121
Waldemar Hołubowski, Parabolic subgroups of Vershik-Kerov’s group, Proc. Amer. Math. Soc. 130 (2002), no. 9, 2579–2582. MR 1900864, DOI 10.1090/S0002-9939-02-06397-9
—, Subgroups of unitriangular group of infinite matrices (in preparation).
W. Hołubowski and N. A. Vavilov, Infinite general linear group (in preparation).
G. D. James, The representation theory of the symmetric groups, Lecture Notes in Mathematics, vol. 682, Springer, Berlin, 1978. MR 513828
Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
E. A. Koval′chuk, Groups of growth-bounded permutations, Problems in group theory and homological algebra (Russian), Matematika, Yaroslav. Gos. Univ., Yaroslavl′1988, pp. 18–28 (Russian). MR 1174988
Günter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, Revised edition, Graduate Studies in Mathematics, vol. 22, American Mathematical Society, Providence, RI, 2000. MR 1721834, DOI 10.1090/gsm/022
Yu. G. Leonov, Representation of finitely approximative $2$-groups by infinite unitriangular matrices over a two element field (to appear).
Bruce A. Magurn, An algebraic introduction to $K$-theory, Encyclopedia of Mathematics and its Applications, vol. 87, Cambridge University Press, Cambridge, 2002. MR 1906572, DOI 10.1017/CBO9781107326002
L. Carlitz, A. Wilansky, John Milnor, R. A. Struble, Neal Felsinger, J. M. S. Simoes, E. A. Power, R. E. Shafer, and R. E. Maas, Problems and Solutions: Advanced Problems: 5600-5609, Amer. Math. Monthly 75 (1968), no. 6, 685–687. MR 1534960, DOI 10.2307/2313822
A. S. Oliĭnyk and V. I. Sushchanskiĭ, A free group of infinite unitriangular matrices, Mat. Zametki 67 (2000), no. 3, 382–386 (Russian, with Russian summary); English transl., Math. Notes 67 (2000), no. 3-4, 320–324. MR 1779471, DOI 10.1007/BF02676668
Andrij Olijnyk and Vitaly Sushchansky, Representations of free products by infinite unitriangular matrices over finite fields, Internat. J. Algebra Comput. 14 (2004), no. 5-6, 741–749. International Conference on Semigroups and Groups in honor of the 65th birthday of Prof. John Rhodes. MR 2104778, DOI 10.1142/S0218196704001931
K. C. O’Meara, A new measure of growth for countable-dimensional algebras. II, J. Algebra 172 (1995), no. 1, 214–240. MR 1320631, DOI 10.1006/jabr.1995.1060
W. Sierpiński, Sur une décomposition d’ensembles, Monatsh. Math. Phys. 35 (1928), no. 1, 239–242 (French). MR 1549531, DOI 10.1007/BF01707443
J. Tits, Free subgroups in linear groups, J. Algebra 20 (1972), 250–270. MR 286898, DOI 10.1016/0021-8693(72)90058-0
D. V. Tjukavkin, Rings all of whose one-sided ideals are generated by idempotents, Comm. Algebra 17 (1989), no. 5, 1193–1198. MR 993398, DOI 10.1080/00927878908823783
A. M. Vershik, Asymptotic aspects of the representation theory of symmetric groups [translation of appendix in G. D. James, The representation theory of the symmetric groups, Russian translation, “Mir”, Moscow, 1982; MR0698241 (84c:20002)], Selecta Math. Soviet. 11 (1992), no. 2, 159–179. Selected translations. MR 1166626
A. M. Vershik and S. V. Kerov, On an infinite-dimensional group over a finite field, Funktsional. Anal. i Prilozhen. 32 (1998), no. 3, 3–10, 95 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 32 (1998), no. 3, 147–152 (1999). MR 1659651, DOI 10.1007/BF02463335