On modules over infinite group rings. (English) Zbl 1354.19001
Summary: Let \(R\) be a commutative ring and \(\Gamma\) be an infinite discrete group. The algebraic \(K\)-theory of the group ring \(R[\Gamma]\) is an important object of computation in geometric topology and number theory. When the group ring is Noetherian, there is a companion \(G\)-theory of \(R[\Gamma]\) which is often easier to compute. However, it is exceptionally rare that the group ring is Noetherian for an infinite group. In this paper, we define a version of \(G\)-theory for any finitely generated discrete group. This construction is based on the coarse geometry of the group. It has some expected properties such as independence from the choice of a word metric. We prove that, whenever \(R\) is a regular Noetherian ring of finite global homological dimension and \(\Gamma\) has finite asymptotic dimension and a finite model for the classifying space \(B\Gamma\), the natural Cartan map from the \(K\)-theory of \(R[\Gamma]\) to \(G\)-theory is an equivalence. On the other hand, our \(G\)-theory is indeed better suited for computation as we show in a separate paper. Some results and constructions in this paper might be of independent interest as we learn to construct projective resolutions of finite type for certain modules over group rings.
MSC:
| 19D50 | Computations of higher \(K\)-theory of rings |
| 20F69 | Asymptotic properties of groups |
| 20J05 | Homological methods in group theory |
| 16E10 | Homological dimension in associative algebras |
| 16E20 | Grothendieck groups, \(K\)-theory, etc. |
| 16S34 | Group rings |
References:
| [1] | 1. G. Bell, Asymptotic properties of groups acting on complexes, Proc. Amer. Math. Soc.133 (2005) 387-396. genRefLink(16, ’S0218196716500181BIB001’, ’10.1090 · Zbl 1133.20030 |
| [2] | 2. G. Bell and A. Dranishnikov, Asymptotic dimension in Bedlewo, Topology Proc.38 (2011) 209-236. · Zbl 1261.20044 |
| [3] | 3. G. Bell and A. Dranishnikov, A Hurewicz-type theorem for asymptotic dimension and applications to geometric group theory, Trans. Amer. Math. Soc.358 (2006) 4749-4764. genRefLink(16, ’S0218196716500181BIB003’, ’10.1090 · Zbl 1117.20032 |
| [4] | 4. P. H. Berridge and M. J. Dunwoody, Non-free projective modules for torsion-free groups, J. London Math. Soc.19 (1979) 193-215. genRefLink(128, ’S0218196716500181BIB004’, ’A1979HA97500001’); · Zbl 0399.20005 |
| [5] | 5. M. Bestvina, K. Bromberg and K. Fujiwara, Constructing group actions on quasi-trees and applications to mapping class groups, preprint (2010), arXiv:1006.1939. · Zbl 1372.20029 |
| [6] | 6. G. Carlsson, Bounded K-theory and the assembly map in algebraic K-theory, in Novikov Conjectures, Index Theory and Rigidity, Vol. 2, eds. S. C. Ferry, A. Ranicki and J. Rosenberg (Cambridge University Press, Cambridge, 1995), pp. 5-127. genRefLink(16, ’S0218196716500181BIB006’, ’10.1017 |
| [7] | 7. G. Carlsson and B. Goldfarb, On homological coherence of discrete groups, J. Algebra276 (2004) 502-514. genRefLink(16, ’S0218196716500181BIB007’, ’10.1016 |
| [8] | 8. G. Carlsson and B. Goldfarb, The integral K-theoretic Novikov conjecture for groups with finite asymptotic dimension, Invent. Math.157 (2004) 405-418. genRefLink(16, ’S0218196716500181BIB008’, ’10.1007 |
| [9] | 9. G. Carlsson and B. Goldfarb, Controlled algebraic G-theory, I, J. Homotopy Relat. Struct.6 (2011) 119-159. genRefLink(128, ’S0218196716500181BIB009’, ’000305945200008’); |
| [10] | 10. G. Carlsson and B. Goldfarb, Algebraic K-theory of geometric groups, preprint (2013), arXiv:1305.3349. |
| [11] | 11. G. Carlsson and B. Goldfarb, Equivariant stable homotopy methods in the algebraic K-theory of infinite groups, preprint (2014), arXiv:1412.3483. |
| [12] | 12. G. Carlsson and B. Goldfarb, K-theory with fibered control, preprint (2014), arXiv:1404.5606. |
| [13] | 13. A. Dranishnikov, On hypersphericity of manifolds with finite asymptotic dimension, Trans. Amer. Math. Soc.355 (2003) 155-167. genRefLink(16, ’S0218196716500181BIB013’, ’10.1090 |
| [14] | 14. A. Dranishnikov, Asymptotic topology, Russian Math. Surveys55 (2000) 1085-1129. genRefLink(16, ’S0218196716500181BIB014’, ’10.1070 |
| [15] | 15. A. Dranishnikov and T. Januszkiewicz, Every Coxeter group acts amenably on a compact space, in Proc. Spring Topology and Dynamics Conf., University of Utah, March 18-20, Vol. 24, Topology Proc. (1999). · Zbl 0973.20029 |
| [16] | 16. A. Dranishnikov and M. Sapir, On the dimension growth of groups, J. Algebra347 (2011) 23-39. genRefLink(16, ’S0218196716500181BIB016’, ’10.1016 |
| [17] | 17. A. Dranishnikov and J. Smith, Asymptotic dimension of discrete groups, Fund. Math.189 (2006) 27-34. genRefLink(16, ’S0218196716500181BIB017’, ’10.4064 |
| [18] | 18. M. J. Dunwoody, Relation modules, Bull. London Math. Soc.3 (1972) 151-155. genRefLink(16, ’S0218196716500181BIB018’, ’10.1112 |
| [19] | 19. B. Goldfarb and T. K. Lance, Controlled algebraic G-theory, II, Contemp. Math.620 (2014) 111-132. genRefLink(16, ’S0218196716500181BIB019’, ’10.1090 · Zbl 1362.19002 |
| [20] | 20. M. Gromov, Asymptotic invariants of infinite groups, in Geometric Groups Theory, Vol. 2 (Cambridge University Press, Cambridge, 1993). · Zbl 0841.20039 |
| [21] | 21. E. Guenther, R. Tessera and G. Yu, A notion of geometric complexity and its application to topological rigidity, preprint (2010), arXiv:1008.0884. |
| [22] | 22. J. Harlander and J. A. Jensen, Exotic relation modules and homotopy types for certain 1-relator groups, Algebr. Geom. Top.6 (2006) 2163-2173. genRefLink(16, ’S0218196716500181BIB022’, ’10.2140 |
| [23] | 23. L. Ji, Asymptotic dimension and the integral K-theoretic Novikov conjecture for arithmetic groups, J. Differential Geom.68 (2004) 535-544. genRefLink(128, ’S0218196716500181BIB023’, ’000230836900004’); · Zbl 1079.55012 |
| [24] | 24. L. Ji, Integral Novikov conjectures for S-arithmetic groups I, K-theory38 (2007) 35-47. genRefLink(16, ’S0218196716500181BIB024’, ’10.1007 |
| [25] | 25. L. Ji, The integral Novikov conjectures for linear groups containing torsion elements, J. Topology1 (2008) 306-316. genRefLink(16, ’S0218196716500181BIB025’, ’10.1112 |
| [26] | 26. F. E. A. Johnson, Syzygies and Homotopy Theory (Springer-Verlag, Berlin, 2012). genRefLink(16, ’S0218196716500181BIB026’, ’10.1007 · Zbl 1233.13001 |
| [27] | 27. P. H. Kropholler, Modules possessing projective resolutions of finite type, J. Algebra216 (1999) 40-55. genRefLink(16, ’S0218196716500181BIB027’, ’10.1006 |
| [28] | 28. W. Lück, Dimension theory of arbitrary modules over finite von Neumann algebras and L2-Betti numbers II: Applications to Grothendieck groups, L2-Euler characteristics and Burnside groups, J. Reine Angew. Math.496 (1998) 213-236. genRefLink(16, ’S0218196716500181BIB028’, ’10.1515 · Zbl 1001.55019 |
| [29] | 29. D. Osin, Asymptotic dimension of relatively hyperbolic groups, Int. Math Res. Not.35 (2005) 2143-2161. genRefLink(16, ’S0218196716500181BIB029’, ’10.1155 · Zbl 1089.20028 |
| [30] | 30. E. K. Pedersen and C. Weibel, A nonconnective delooping of algebraic K-theory, in Algebraic and Geometric Topology, (eds.) A. Ranicki, N. Levitt and F. Quinn, Lecture Notes in Mathematics, Vol. 1126 (Springer-Verlag, Berlin, 1985), pp. 166-181. genRefLink(16, ’S0218196716500181BIB030’, ’10.1007 · Zbl 0591.55002 |
| [31] | 31. J. Roe, Hyperbolic groups have finite asymptotic dimension, Proc. Amer. Math. Soc.133 (2005) 2489-2490. genRefLink(16, ’S0218196716500181BIB031’, ’10.1090 |
| [32] | 32. R. W. Thomason and T. Trobaugh, Higher algebraic K-theory of schemes and of derived categories, in The Grothendieck Festschrift, Vol. III, Progress in Mathematics, Vol. 88 (Birkhäuser, Basel, 1990), pp. 247-435. genRefLink(16, ’S0218196716500181BIB032’, ’10.1007 · Zbl 0731.14001 |
| [33] | 33. D. Quillen, Higher algebraic K-theory: I, in Algebraic K-theory I, ed. H. Bass, Lecture Notes in Mathematics, Vol. 341 (Springer-Verlag, Berlin, 1973), pp. 77-139. genRefLink(16, ’S0218196716500181BIB033’, ’10.1007 · Zbl 0292.18004 |
| [34] | 34. N. Wright, Finite asymptotic dimension for CAT(0) cube complexes, Geom. Topol.16 (2012) 527-554. genRefLink(16, ’S0218196716500181BIB034’, ’10.2140 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
