\(L^2\)-Betti numbers and computability of reals. (English) Zbl 07737094
Summary: We study the computability degree of real numbers arising as \(L^2\)-Betti numbers or \(L^2\)-torsion of groups, parametrised over the Turing degree of the word problem.
MSC:
| 03Dxx | Computability and recursion theory |
Software:
mathlibReferences:
| [1] | M.F. Atiyah, Elliptic operators, discrete groups and von Neumann algebras, in: Colloque “Analyse et Topolo-gie” en l’honneur de Henri Cartan, Société Mathématique de France (SMF), 1976, pp. 43-72. · Zbl 0323.58015 |
| [2] | T. Austin, Rational group ring elements with kernels having irrational dimension, Proc. Lond. Math. Soc. (3) 107(6) (2013), 1424-1448. doi:10.1112/plms/pdt029. · Zbl 1295.20041 · doi:10.1112/plms/pdt029 |
| [3] | J. Avigad, L. de Moura and S. Kong, Theorem proving in Lean, 2021. Release 3.23.0, https:// leanprover.github.io/theorem_proving_in_lean/. |
| [4] | W.E. Egbert, Personal calculator algorithms IV: Logarithmic functions, Hewlett-Packard Journal 4 (1978), 29-32. |
| [5] | G. Elek and E. Szabó, Hyperlinearity, essentially free actions and L 2 -invariants. The sofic property, Math. Ann. 332(2) (2005), 421-441. doi:10.1007/s00208-005-0640-8. · Zbl 1070.43002 · doi:10.1007/s00208-005-0640-8 |
| [6] | G. Elek and E. Szabó, On sofic groups, J. Group Theory 9(2) (2006), 161-171. doi:10.1515/JGT.2006. 011. · Zbl 1153.20040 · doi:10.1515/JGT.2006.011 |
| [7] | F. Fournier-Facio, C. Löh and M. Moraschini, Bounded cohomology of finitely presented groups: Vanishing, non-vanishing, and computability, Annali della Scuola Normale Superiore di Pisa (2021), to appear. |
| [8] | Ł. Grabowski, On Turing dynamical systems and the Atiyah problem, Invent. Math. 198(1) (2014), 27-69. doi:10.1007/s00222-013-0497-5. · Zbl 1354.20026 · doi:10.1007/s00222-013-0497-5 |
| [9] | Ł. Grabowski, Vanishing of l 2 -cohomology as a computational problem, Bull. Lond. Math. Soc. 47(2) (2015), 233-247. doi:10.1112/blms/bdu114. · Zbl 1433.57010 · doi:10.1112/blms/bdu114 |
| [10] | P.A. Griffith, Infinite Abelian Group Theory, University of Chicago Press, Chicago, Ill.-London, 1970. · Zbl 0204.35001 |
| [11] | T. Groth, l 2 -Bettizahlen endlich präsentierter Gruppen, BSc thesis, Universität Göttingen, 2012. |
| [12] | N. Heuer, The full spectrum of scl on recursively presented groups, Geometricae Dedicata (2019), to appear. |
| [13] | N. Heuer and C. Löh, Transcendental simplicial volumes, Annales de l’Institut Fourier (2021), to appear. |
| [14] | H. Kammeyer, Introduction to 2 -Invariants, Springer, 2019. · Zbl 1458.55001 |
| [15] | R.C. Kirby and L.C. Siebenmann, On the triangulation of manifolds and the Hauptvermutung, Bull. Amer. Math. Soc. 75 (1969), 742-749. doi:10.1090/S0002-9904-1969-12271-8. · Zbl 0189.54701 · doi:10.1090/S0002-9904-1969-12271-8 |
| [16] | P.A. Linnell, Division rings and group von Neumann algebras, Forum Math. 5(6) (1993), 561-576. · Zbl 0794.22008 |
| [17] | C. Löh and M. Uschold, L 2 -Betti numbers and computability of reals -Lean project. https://gitlab. com/L2-comp/l2-comp-lean, 2022. |
| [18] | C. Löh and M. Uschold, L 2 -Betti numbers and computability of reals, 2022, arXiv:2202.03159v2. |
| [19] | W. Lück, Approximating L 2 -invariants by their finite-dimensional analogues, Geom. Funct. Anal. 4(4) (1994), 455-481. doi:10.1007/BF01896404. · Zbl 0853.57021 · doi:10.1007/BF01896404 |
| [20] | W. Lück, L 2 -Invariants: Theory and Applications to Geometry and K-Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], Vol. 44, Springer, 2002. · Zbl 1009.55001 |
| [21] | C.F. Miller III, Decision problems for groups -survey and reflections, in: Algorithms and Classification in Combinatorial Group Theory. Lectures of a Workshop on Algorithms, Word Problems and Classification in Combinatorial Group Theory, Held at MSRI, Berkeley, CA, USA, January 1989, Springer-Verlag, 1989, pp. 1-59. · Zbl 0752.20014 |
| [22] | A. Nabutovsky and S. Weinberger, Betti numbers of finitely presented groups and very rapidly growing func-tions, Topology 46(2) (2007), 211-223. doi:10.1016/j.top.2007.02.002. · Zbl 1139.20030 · doi:10.1016/j.top.2007.02.002 |
| [23] | C. Neoftyidis, S. Wang and Z. Wang, Realising sets of integers as mapping degree sets, Bulletin of the London Mathematical Society (2023), doi:10.1112/blms.12813. · Zbl 1529.55002 · doi:10.1112/blms.12813 |
| [24] | B.H. Neumann and H. Neumann, Embedding theorems for groups, Journal of the London Mathematical Soci-ety s1-34(4) (1959), 465-479. doi:10.1112/jlms/s1-34.4.465. · Zbl 0102.26401 · doi:10.1112/jlms/s1-34.4.465 |
| [25] | M. Pichot, T. Schick and A. Zuk, Closed manifolds with transcendental L 2 -Betti numbers, J. Lond. Math. Soc., II. Ser. 92(2) (2015), 371-392. doi:10.1112/jlms/jdv026. · Zbl 1347.20045 · doi:10.1112/jlms/jdv026 |
| [26] | R. Rettinger and X. Zheng, Computability of real numbers, in: Handbook of Computability and Complexity in Analysis, Springer, Cham, 2021, pp. 3-28. doi:10.1007/978-3-030-59234-9_1. · Zbl 07464639 · doi:10.1007/978-3-030-59234-9_1 |
| [27] | J.J. Rotman, An Introduction to the Theory of Groups, Graduate Texts in Mathematics, Vol. 148, Springer-Verlag, 1995. · Zbl 0810.20001 |
| [28] | T. Schick, L 2 -Determinant class and approximation of L 2 -Betti numbers, Trans. Am. Math. Soc. 353(8) (2001), 3247-3265. doi:10.1090/S0002-9947-01-02699-X. · Zbl 0979.55004 · doi:10.1090/S0002-9947-01-02699-X |
| [29] | L.C. Siebenmann, On the homotopy type of compact topological manifolds, Bull. Amer. Math. Soc. 74 (1968), 738-742. doi:10.1090/S0002-9904-1968-12022-1. · Zbl 0165.56703 · doi:10.1090/S0002-9904-1968-12022-1 |
| [30] | S.G. Simpson, Degrees of unsolvability: A survey of results, Barewise, Jon, 1977, pp. 631-652. |
| [31] | The mathlib Community. The Lean Mathematical Library. Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs (CPP ’2), 2020. |
| [32] | X. Zheng, On the Turing degrees of weakly computable real numbers, J. Log. Comput. 13(2) (2003), 159-172. doi:10.1093/logcom/13.2.159. · Zbl 1054.03040 · doi:10.1093/logcom/13.2.159 |
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