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L2-Betti Numbers and their Analogues in Positive Characteristic

Published online by Cambridge University Press:  15 April 2019

By
Andrei Jaikin-Zapirain
Edited by
C. M. Campbell ,
C. W. Parker ,
M. R. Quick ,
E. F. Robertson  and
C. M. Roney-Dougal
C. M. Campbell
Affiliation:
University of St Andrews, Scotland
C. W. Parker
Affiliation:
University of Birmingham
M. R. Quick
Affiliation:
University of St Andrews, Scotland
E. F. Robertson
Affiliation:
University of St Andrews, Scotland
C. M. Roney-Dougal
Affiliation:
University of St Andrews, Scotland
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Summary

In this article, we give a survey of results on L2-Betti numbers and their analogues in positive characteristic. The main emphasis is made on the Lück approximation conjecture and the strong Atiyah conjecture.

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Chapter
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Groups St Andrews 2017 in Birmingham , pp. 346 - 405
Publisher: Cambridge University Press
Print publication year: 2019

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References

Abért, M., Jaikin-Zapirain, A. and Nikolov, N., The rank gradient from a combinatorial viewpoint, Groups Geom. Dyn. 5 (2011), 213–230.CrossRefGoogle Scholar
Abért, M. and Nikolov, N., Rank gradient, cost of groups and the rank versus Heegaard genus problem, J. Eur. Math. Soc. 14 (2012), 1657–1677.CrossRefGoogle Scholar
Abért, M., Thom, A. and Virag, B., Benjamini-Schramm convergence and pointwise convergence of the spectral measure, preprint, www.renyi.hu/∼abert/luckapprox.pdf.Google Scholar
Agol, I., The virtual Haken conjecture. With an appendix by Agol, Groves, D., and Manning, J., Doc. Math. 18 (2013), 1045–1087.Google Scholar
Antolin, Y. and Jaikin-Zapirain, A., The Hanna Neumann conjecture for hyperbolic limit groups, in preparation.Google Scholar
Ara, P. and Claramunt, J., Uniqueness of the von Neumann continuous factor, arXiv preprint, arXiv:1705.04501 (2017).Google Scholar
Ara, P. and Goodearl, K. R., The realization problem for some wild monoids and the Atiyah problem, Trans. Amer. Math. Soc. 369 (2017), 5665–5710.Google Scholar
Ardakov, K. and Brown, K., Primeness, semiprimeness and localisation in Iwasawa algebras, Trans. Amer. Math. Soc. 359 (2007), 1499–1515.Google Scholar
Atiyah, M. F., Elliptic operators, discrete groups and von Neumann algebras, Colloque “Analyse et Topologie” en l’Honneur de Henri Cartan (Orsay, 1974), Astérisque 32–33 (Soc. Math. France, Paris 1976), 43–72.Google Scholar
Austin, T., Rational group ring elements with kernels having irrational dimension, Proc. Lond. Math. Soc. (3) 107 (2013), 1424–1448.CrossRefGoogle Scholar
Bartholdi, L., Amenability of groups is characterized by Myhill’s Theorem, with an appendix by Kielak, D., J. Eur. Math. Soc., to appear.Google Scholar
Berberian, S. K., Baer ∗-rings, Die Grundlehren der Mathematischen Wissenschaften, Band 195 (Springer-Verlag, New York-Berlin, 1972).CrossRefGoogle Scholar
Berberian, S. K., The maximal ring of quotients of a finite von Neumann algebra, Rocky Mountain J. Math. 12 (1982), 149–164.CrossRefGoogle Scholar
Bergeron, N., Linnell, P., Lück, W. and Sauer, R., On the growth of Betti numbers in p-adic analytic towers, Groups Geom. Dyn. 8 (2014), 311–329.CrossRefGoogle Scholar
Blomer, I., Linnell, P. and Schick, T., Galois cohomology of completed link groups, Proc. Amer. Math. Soc. 136 (2008), 3449–3459.CrossRefGoogle Scholar
Calegari, F. and Emerton, M., Mod-p cohomology growth in p-adic analytic towers of 3-manifolds, Groups Geom. Dyn. 5 (2011), 355–366.CrossRefGoogle Scholar
Calegari, F. and Emerton, M., Completed cohomology – a survey, Non-abelian fundamental groups and Iwasawa theory (London Math. Soc. Lecture Note Ser., 393, Cambridge Univ. Press, Cambridge, 2012), 239–257.Google Scholar
Cheeger, J. and Gromov, M., L2-cohomology and group cohomology, Topology 25 (1986), 189–215.CrossRefGoogle Scholar
Cohn, P. M., Skew fields. Theory of general division rings, Encyclopedia of Mathematics and its Applications 57 (Cambridge University Press, Cambridge, 1995).CrossRefGoogle Scholar
Cohn, P. M., Free ideal rings and localization in general rings, New Mathematical Monographs 3 (Cambridge University Press, Cambridge, 2006).CrossRefGoogle Scholar
Dicks, W., Mineyev, Simplified, preprint, http://mat.uab.es/∼dicks.Google Scholar
Dicks, W. and Schick, T., The spectral measure of certain elements of the complex group ring of a wreath product, Geom. Dedicata 93 (2002), 121–137.CrossRefGoogle Scholar
Dixon, J., du Sautoy, M., Mann, A. and Segal, D., Analytic pro-p groups, Second edition, Cambridge Studies in Advanced Mathematics 61 (Cambridge University Press, Cambridge 1999).CrossRefGoogle Scholar
Dodziuk, J., de Rham-Hodge theory for L2-cohomology of infinite coverings, Topology 16 (1977), 157–165.CrossRefGoogle Scholar
Dodziuk, J. and Mathai, V., Approximating L2-invariants of amenable covering spaces: A combinatorial approach, J. Funct. Anal. 154 (1998), 359–378.CrossRefGoogle Scholar
Dodziuk, J., Linnell, P., Mathai, V., Schick, T. and Yates, S., Approximating L2invariants and the Atiyah conjecture. Dedicated to the memory of Jürgen K. Moser, Comm. Pure Appl. Math. 56 (2003), 839–873.CrossRefGoogle Scholar
Dudley, R. M., Real analysis and probability, Revised reprint of the 1989 original, Cambridge Studies in Advanced Mathematics 74 (Cambridge University Press, Cambridge 2002).Google Scholar
Dykema, K., Heister, T. and Juschenko, K., Finitely presented groups related to Kaplansky’s direct finiteness conjecture, Exp. Math. 24 (2015), 326–338.CrossRefGoogle Scholar
Eckmann, B., Introduction to l2-methods in topology: reduced l2-homology, harmonic chains, l2-Betti numbers. Notes prepared by Guido Mislin, Israel J. Math. 117 (2000), 183–219.CrossRefGoogle Scholar
Elek, G., The rank of finitely generated modules over group algebras, Proc. Amer. Math. Soc. 131 (2003), 3477–3485.CrossRefGoogle Scholar
Elek, G., The strong approximation conjecture holds for amenable groups, J. Funct. Anal. 239 (2006), 345–355.CrossRefGoogle Scholar
Elek, G., Connes embeddings and von Neumann regular closures of amenable group algebras, Trans. Amer. Math. Soc. 365 (2013), 3019–3039.Google Scholar
Elek, G., Lamplighter groups and von Neumann’s continuous regular ring, Proc. Amer. Math. Soc. 144 (2016), 2871–2883.CrossRefGoogle Scholar
Elek, G. and Szabó, E., Sofic groups and direct finiteness, J. Algebra 280 (2004), 426–434.CrossRefGoogle Scholar
Elek, G. and Szabó, E., Hyperlinearity, essentially free actions and L2-invariants. The sofic property, Math. Ann. 332 (2005), 421–441.CrossRefGoogle Scholar
Elek, G. and Szabó, E., On sofic groups, J. Group Theory 9 (2006), 161–171.CrossRefGoogle Scholar
Elek, G. and Szabó, E., Sofic representations of amenable groups, Proc. Amer. Math. Soc. 139 (2011), 4285–4291.CrossRefGoogle Scholar
Ershov, M. and Lück, W., The first L2-Betti number and approximation in arbitrary characteristic, Doc. Math. 19 (2014), 313–332.CrossRefGoogle Scholar
Farkas, D. and Linnell, P., Congruence subgroups and the Atiyah conjecture, Groups, rings and algebras, Contemp. Math. 420 (Amer. Math. Soc., Providence, RI 2006), 89–102.Google Scholar
Friedman, J., Sheaves on graphs, their homological invariants, and a proof of the Hanna Neumann conjecture: with an appendix by Warren Dicks, Mem. Amer. Math. Soc. 233 (2014).Google Scholar
Gildenhuys, D., On pro-p-groups with a single defining relator, Invent. Math. 5 (1968), 357–366.CrossRefGoogle Scholar
Grabowski, L., On Turing dynamical systems and the Atiyah problem, Invent. Math. 198 (2014), 27–69.CrossRefGoogle Scholar
Grabowski, L., Group ring elements with large spectral density, Math. Ann. 363 (2015), 637–656.CrossRefGoogle Scholar
Grabowski, L., Irrational l2-invariants arising from the lamplighter group, Groups Geom. Dyn. 10 (2016), 795–817.CrossRefGoogle Scholar
Grabowski, L. and Schick, T., On computing homology gradients over finite fields, Math. Proc. Cambridge Philos. Soc. 162 (2017), 507–532.CrossRefGoogle Scholar
Goodearl, K. R., von Neumann regular rings, Second edition, (Robert E. Krieger Publishing Co., Inc., Malabar, FL 1991).Google Scholar
Goodearl, K. R. and Warfield, R. B., An introduction to noncommutative Noetherian rings, Second edition, London Math. Soc. Student Texts 61 (Cambridge University Press, Cambridge 2004).CrossRefGoogle Scholar
Grigorchuk, R., Linnell, P., Schick, T. and Zuk, A., On a question of Atiyah, C. R. Acad. Sci. Ser I Math. 331 (2000), 663–668.Google Scholar
Grunewald, F., Jaikin-Zapirain, A., Pinto, A. and Zalesskii, P., Normal subgroups of profinite groups of non-negative deficiency, J. Pure Appl. Algebra 218 (2014), 804–828.CrossRefGoogle Scholar
Grigorchuk, R. and Zuk, A., The lamplighter group as a group generated by a 2-state automation, and its spectrum, Geom. Dedicata 87 (2001), 209–244.CrossRefGoogle Scholar
Gromov, M., Kähler hyperbolicity and L2-Hodge theory, J. Differential Geom. 33 (1991), 263–292.CrossRefGoogle Scholar
Harris, M., p-adic representations arising from descent on abelian varieties, Compositio Math. 39 (1979), 177–245,Google ScholarGoogle Scholar
Hayes, B. and Sale, A., Metric approximations of wreath products, to appear in Ann. Inst. Fourier.Google Scholar
Howson, A. G., On the intersection of finitely generated free groups, J. London Math. Soc. 29 (1954), 428–434.Google Scholar
Jaikin-Zapirain, A., Approximation by subgroups of finite index and the Hanna Neumann conjecture, Duke Math. J. 166 (2017), 1955–1987.CrossRefGoogle Scholar
Jaikin-Zapirain, A., The base change in the Atiyah and Lück approximation conjectures, preprint, http://verso.mat.uam.es/∼andrei.jaikin/preprints/sac.pdf.Google Scholar
Jaikin-Zapirain, A. and López-Álvarez, D., On the space of Sylvester matrix rank functions, in preparation.Google Scholar
Jaikin-Zapirain, A. and Shusterman, M., The Hanna Neumann conjecture for Demushkin groups, in preparation.Google Scholar
Kammeyer, H., Introduction to L2-invariants, course notes available online at https://topology.math.kit.edu/21679.php.Google Scholar
Kaplansky, I., Fields and rings, Reprint of the second (1972) edition, Chicago Lectures in Mathematics (University of Chicago Press, Chicago, IL, 1995).Google Scholar
Kionke, S., Characters, L2-Betti numbers and an equivariant approximation theorem, Arxiv:1702.02599.Google Scholar
Kionke, S., The growth of Betti numbers and approximation theorems, Arxiv:1709.00769.Google Scholar
Knebusch, A., Linnell, P. and Schick, T., On the center-valued Atiyah conjecture for L2-Betti numbers, Doc. Math. 22 (2017), 659–677.CrossRefGoogle Scholar
Kowalski, E., Spectral theory in Hilbert spaces, online notes www.math.ethz.ch/∼kowalski/spectral-theory.pdf.Google Scholar
Kropholler, P., Linnell, P. and Moody, J., Applications of a new K-theoretic theorem to soluble group rings, Proc. Amer. Math. Soc. 104 (1988), 675–684.Google Scholar
Lazard, M., Groupes analytiques p-adiques, Inst. Hautes Études Sci. Publ. Math. 26 (1965).Google Scholar
Lehner, F. and Wagner, S., Free lamplighter groups and a question of Atiyah, Amer. J. Math. 135 (2013), 835–849.CrossRefGoogle Scholar
Linnell, P., Division rings and group von Neumann algebras, Forum Math. 5 (1993), 561–576.CrossRefGoogle Scholar
Linnell, P., Lück, W. and Sauer, R., The limit of Fp-Betti numbers of a tower of finite covers with amenable fundamental groups, Proc. Amer. Math. Soc. 139 (2011), 421–434.CrossRefGoogle Scholar
Linnell, P., Okun, B. and Schick, T., The strong Atiyah conjecture for right-angled Artin and Coxeter groups, Geom. Dedicata 158 (2012), 261–266.CrossRefGoogle Scholar
Linnell, P. and Schick, T., Finite group extensions and the Atiyah conjecture, J. Amer. Math. Soc. 20 (2007), 1003–1051.CrossRefGoogle Scholar
Linnell, P. and Schick, T., The Atiyah conjecture and Artinian rings, Pure Appl. Math. Q. 8 (2012), 313–327.CrossRefGoogle Scholar
Lott, J., Heat kernels on covering spaces and topological invariants, J. Differential Geom. 35 (1992), 471–510.CrossRefGoogle Scholar
Lubotzky, A., Combinatorial group theory for pro-p groups, J. Pure Appl. Algebra 25 (1982), 311–325.CrossRefGoogle Scholar
Lubotzky, A., A group theoretic characterization of linear groups, J. Algebra 113 (1988), 207–214.CrossRefGoogle Scholar
Lück, W., Approximating L2-invariants by their finite-dimensional analogues, Geom. Funct. Anal. 4 (1994), 455–481.CrossRefGoogle Scholar
Lück, W., The relation between the Baum-Connes conjecture and the trace conjecture, Invent. Math. 149 (2002), 123–152.CrossRefGoogle Scholar
Lück, W., L2-invariants: theory and applications to geometry and K-theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics 44 (Springer-Verlag, Berlin, 2002).CrossRefGoogle Scholar
Lück, W., L2-invariants from the algebraic point of view, Geometric and cohomological methods in group theory, London Math. Soc. Lecture Note Ser. 358 (Cambridge Univ. Press, Cambridge, 2009), 63–161.Google Scholar
Lück, W., Estimates for spectral density functions of matrices over C[Zd], Ann. Math. Blaise Pascal 22 (2015), 73–88.CrossRefGoogle Scholar
Lück, W., Approximating L2-invariants by their classical counterparts, EMS Surv. Math. Sci. 3 (2016), 269–344.CrossRefGoogle Scholar
Lyndon, R., Cohomology theory of groups with a single defining relation, Ann. of Math. (2) 52 (1950), 650–665.CrossRefGoogle Scholar
Malcolmson, P., Determining homomorphisms to skew fields, J. Algebra 64 (1980), 399–413.CrossRefGoogle Scholar
McConnell, J. C. and Robson, J. C., Noncommutative Noetherian rings. With the cooperation of L. W. Small. Revised edition, Graduate Studies in Mathematics 30 (American Mathematical Society, Providence, RI 2001).Google Scholar
Mineyev, I., Submultiplicativity and the Hanna Neumann conjecture, Ann. of Math. (2) 175 (2012), 393–414.CrossRefGoogle Scholar
Moody, J., Brauer induction for G0 of certain infinite groups, J. Algebra 122 (1989), 1–14.CrossRefGoogle Scholar
Neumann, A., Completed group algebras without zero divisors, Arch. Math. 51 (1988), 496–499.CrossRefGoogle Scholar
Neumann, H., On the intersection of finitely generated free groups, Publ. Math. Debrecen 4 (1956), 186–189;Google ScholarGoogle Scholar
Neumann, W. D., On intersections of finitely generated subgroups of free groups, Groups–Canberra 1989 (Lecture Notes in Math., 1456, Springer, Berlin, 1990), 161–170.Google Scholar
Northcott, D. and Reufel, M., A generalization of the concept of length, Quart. J. Math. Oxford Ser. (2) 16 (1965), 297–321.CrossRefGoogle Scholar
Ornstein, D. and Weiss, B., Entropy and isomorphism theorems for actions of amenable groups, J. Anal. Math. 48 (1987), 1–141.CrossRefGoogle Scholar
Osin, D., L2-Betti numbers and non-unitarizable groups without free subgroups, Int. Math. Res. Not. IMRN (2009), 4220–4231.CrossRefGoogle Scholar
Osin, D., On acylindrical hyperbolicity of groups with positive first l2-Betti number, Bull. Lond. Math. Soc. 47 (2015), 725–730.CrossRefGoogle Scholar
Pansu, P., Introduction to L2 Betti numbers, Riemannian geometry (Waterloo, ON, 1993), Fields Inst. Monogr., 4 (Amer. Math. Soc., Providence, RI, 1996), 53–86.Google Scholar
Pape, D., A short proof of the approximation conjecture for amenable groups, J. Funct. Anal. 255 (2008), 1102–1106.CrossRefGoogle Scholar
Petersen, H. D., Sauer, R. and Thom, A., L2-Betti numbers of totally disconnected groups and their approximation by Betti numbers of lattices, Arxiv:1612.04559.Google Scholar
Pichot, M., Schick, T., and Zuk, A., Closed manifolds with transcendental L2-Betti numbers, J. London Math. Soc. 92 (2015), 371–392.CrossRefGoogle Scholar
Reich, H., Group von Neumann Algebras and Algebras, Related, Ph.D. Thesis (Göttingen 1998), www.mi.fu-berlin.de/math/groups/top/members/publ/diss.pdf.Google Scholar
Rowen, L. and Saltman, D., Tensor products of division algebras and fields, J. Algebra 394 (2013), 296–309.CrossRefGoogle Scholar
Salce, L. and Virili, S., The addition theorem for algebraic entropies induced by nondiscrete length functions, Forum Math. 28 (2016), 1143–1157.CrossRefGoogle Scholar
Sauer, R., Power series over the group ring of a free group and applications to Novikov-Shubin invariants, High-dimensional manifold topology (World Sci. Publ., River Edge, NJ 2003), 449–468.Google Scholar
Sauer, R. and Thom, A., A spectral sequence to compute L2-Betti numbers of groups and groupoids, J. Lond. Math. Soc. (2) 81 (2010), 747–773.CrossRefGoogle Scholar
Schick, T., Integrality of L2-Betti numbers, Math. Ann. 317 (2000), 727–750,Google ScholarGoogle Scholar
Schick, T., L2-determinant class and approximation of L2-Betti numbers, Trans. Amer. Math. Soc. 353 (2001), 3247–3265.CrossRefGoogle Scholar
Schofield, A., Representation of rings over skew fields, London Math. Soc. Lecture Note Ser. 92 (Cambridge University Press, Cambridge, 1985).CrossRefGoogle Scholar
Schreve, K., The strong Atiyah conjecture for virtually cocompact special groups, Math. Ann. 359 (2014), 629–636.CrossRefGoogle Scholar
Stenström, B., Rings of quotients, Die Grundlehren der Mathematischen Wissenschaften, Band 217. An introduction to methods of ring theory (Springer-Verlag, New York-Heidelberg, 1975).CrossRefGoogle Scholar
Symonds, P. and Weigel, T., Cohomology of p-adic analytic groups, New horizons in pro-p groups (Progr. Math., 184, Birkhäuser Boston, Boston, MA, 2000), 349–410.Google Scholar
Szoke, N. G., Sofic groups, M.Sc. Thesis (Eötvös Loránd University 2014). www.cs.elte.hu/blobs/diplomamunkak/msc mat/2014/szoke nora gabriella.pdfGoogle Scholar
Thom, A., Sofic groups and Diophantine approximation, Comm. Pure Appl. Math. 61 (2008), 1155–1171.CrossRefGoogle Scholar
Vámos, P., Additive functions and duality over Noetherian rings, Quart. J. Math. Oxford Ser. (2) 19 (1968), 43–55.CrossRefGoogle Scholar
Virili, S., Crawley-Boewey’s extension of invariants in the non-discrete case, preprint 2017.Google Scholar
Virili, S., Algebraic entropy of amenable group actions, preprint 2017.Google Scholar
Wise, D., Research announcement: the structure of groups with a quasiconvex hierarchy, Electron. Res. Announc. Math. Sci. 16 (2009), 44–55.Google Scholar

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