Publication: Bredon homology of artin groups of dihedral type
Loading...
Official URL
Full text at PDC
Publication Date
2022
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Citations
Exportar
Abstract
For Artin groups of dihedral type, we compute the Bredon homology groups of theclassifying space for the family of virtually cyclic subgroups with coefficients in the K-theory of a group ring.
Description
UCM subjects
Unesco subjects
Keywords
Citation
[1] J. Aramayona, D. Juan-Pineda and A. Trujillo-Negrete, On the virtually cyclic dimension of mapping class groups of punctured spheres, Algebr. Geom. Topol. 18 (2018), no. 4, 2471—2495.
[2] J. Aramayona and C. Martínez-Pérez, The proper geometric dimension of the mapping class group, Algebr. Geom. Topol.14 (2014), no. 1, 217-–227.
[3] G.N. Arzhantseva, On quasiconvex subgroups of word hyperbolic groups, Geom. Dedicata, 87 (2001), 191–208.
[4] S Azzali, S.L. Browne, M.P. Gomez-Aparicio, L.C. Ruth and H. Wang, K-homology and K-theory of pure braid groups, arXiv:2105.07848
[5] N. Bárcenas, D. Juan-Pineda and M. Velásquez, Bredon cohomology, K-theory and K-homology of pullbacks of groups, arXiv:1311.3138.
[6] H. Bass, Algebraic K-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968.
[7] M. Bestvina. K. Fujiwara and D. Wigglesworth, Farrell-Jones conjecture for free-by-cyclic groupsarXiv:1906.00069.pdf
[8] G.E. Bredon, Equivariant cohomology theories, Lecture Notes in Mathematics, 34. Springer-Verlag, BerlinNew York, 1967.
[9] M. Bustamante and L.J. Sánchez-Saldaña, On the algebraic K-theory of the Hilbert modular group, Algebr.Geom. Topol.16 (2016), 2107-–2125.
[10] D.W. Carter, Lower K-theory of finite groups, Comm. Algebra 8 (1980), 1927-–1937.
[11] P. Cassou-Noguès, Classes d’idéaux de l’algèbre d’un groupe abélien, C. R. Acad. Sci. Paris Ser. A-B,276:A973–A975, 1973.
[12] R. Charney, J. Meier and K. Whitlessey, Bestvina’s normal form complex and the homology of Garside groups, Geom. Dedicata 105 (2004), 171-–188.
[13] L. Ciobanu, D. Holt and S. Rees, Equations in groups that are virtually direct products J. Algebra 545(2020), 88–99.
[14] M. Clancy and G. Ellis, Homology of some Artin and twisted Artin groups, J. K-Theory 6 (2009), 171–196.
[15] J.F. Davis, F. Quinn and H. Reich, Algebraic K-theory over the infinite dihedral group: a controlled topology
approach, J. Topol. 4 (2011), 505-–528.
[16] D. Degrijse, A cohomological characterization of locally virtually cyclic groups, Adv. Math. 305 (2017),935–952.
[17] D. Degrijse and N. Petrosyan, Geometric dimension of groups for the family of virtually cyclic subgroups,J.Topol. 7 (2014), 697–726.
[18] T. tom Dieck, Transformation groups Vol. 8. Walter de Gruyter, 1987.
[19] T. Farrell, The nonfiniteness of Nil, Proc. Amer. Math. Soc. 65 (1977), 215—216.
[20] T. Farrell and L. Jones, Isomorphism conjectures in algebraic K-theory, J. Amer. Math. Soc., 6 (1993),249—297.
[21] R. Flores and J. González-Meneses, Classifying spaces for the family of virtually cyclic subgroups of braid groups, Int. Math. Res. Notices. 2020 (2020), 1575–1600.
[22] J. Guaschi, D. Juan-Pineda and S. Millán-López, The lower algebraic K-theory of virtually cyclic subgroups
of the braid groups of the sphere and of Z[B4(S2)], Springer Briefs in Mathematics. Springer, Cham, 2018.
[23] I. Hambleton and W. Lück, Induction and computation of Bass Nil groups for finite groups, Pure Appl.Math. Q 8(2012), 199–219.
[24] A. Hatcher, Algebraic topology, Cambridge University Press, 2002.
[25] C. Hidber and D. Juan-Pineda, The algebraic K-theory of the group ring of the Klein bottle group, Topol. Appl. 293 (2021), 8 pages.
[26] G. Higman, The units of group rings, Proc. London Math. Soc. 46 (1940), 231–248.
[27] D. Kasprowski, K. Li and W. Lück, K and L-theory of graph products of groups, Groups Geom. Dyn. 15(2021), 269–311.
[28] J. Lafont and I. Ortiz, Relative hyperbolicity, classifying spaces, and lower algebraic K-theory, Topology 46(2007), 527–553.
[29] W. Lück, Survey on classifying spaces for families of subgroups, in Infinite groups: geometric, combinatorial and dynamical aspects, 269—322, Progr. Math., 248, Birkhäuser, Basel, 2005.
[30] W. Lück, Isomorphism conjectures in K and L-theory, in preparation, https://www.him.unibonn.de/lueck/data/ic.pdf
[31] W. Lück and H. Reich, The Baum-Connes and the Farrell-Jones conjectures in K- and L-theory, Handbook of K-theory, volume 2, pp. 703–842, Springer, 2005.
[32] W. Lück and D. Rosenthal, On the K- and L-theory of hyperbolic and virtually finitely generated abelian groups, Forum Math. 26 (2014), no. 5, 1565—1609.
[33] W. Lück and W. Steimle, Splitting the relative assembly map, Nil-terms and involutions, Ann. K-theory 1(2016) 339–377.
[34] W. Lück and M. Weiermann, On the classifying space of the family of virtually cyclic subgroups, Pure Appl.Math. Q. 8 2 (2012), 497–555.
[35] R. Lyndon, Cohomology theory of groups with a single defining relation, Ann. of Math. 52 (1950), 650–665.
[36] D. Macpherson, Permutation groups whose subgroups have just finitely many orbits, in Ordered groups and
infinite permutation groups, pp. 221–230, Kluwer Academic Publishers, 1996.
BREDON HOMOLOGY OF ARTIN GROUPS OF DIHEDRAL TYPE 27
[37] J. Mairesse and F. Mathéus, Growth series for Artin groups of dihedral type Internat. J. Algebra Comput. 16(2006), 1087–1107.
[38] C. Martínez-Pérez, A spectral sequence in Bredon (co)homology, J. Pure Appl. Algebra 176 (2002), 161–173.
[39] G. Mislin and A. Valette, Proper group actions and the Baum-Connes conjecture, Advanced courses in Mathematics, no. 55, CRM Barcelona, 2003.
[40] B.E.A. Nucinkis and N. Petrosyan, Hierarchically cocompact classifying spaces for mapping class groups of
surfaces, Bull. Lond. Math. Soc. 50 (2018), 569–582.
[41] R. Oliver, Whitehead groups of finite groups. Cambridge University Press, Cambridge, 1988.
[42] L. Paris, Braid groups and Artin groups, Handbook of Teichmüller theory. Vol. II, pp. 389—451, IRMA Lect. Math. Theor. Phys., 13, Eur. Math. Soc., Zürich, 2009.
[43] D. Quillen, Finite generation of the groups Ki of rings of algebraic integers, in Algebraic K-theory, I: Higher
K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pages 179–198. Lecture Notes in Math., vol. 341. Springer-Verlag, Berlin, 1973.
[44] R. Sánchez-García, Bredon homology and equivariant K-homology of SL(3, Z), J. Pure Appl. Algebra, 212 (2008), 1046–1059.
[45] L.J. Sánchez-Saldaña and M. Velásquez, The algebraic and topological K-theory of the Hilbert modular group, Homology Homotopy Appl. 20 (2018), no. 2, 377—402.
[46] S.K. Shehgal, Topics in group rings, Dekkar, New York, 1978.
[47] V. Srinivas, Algebraic K-theory, Birkhäuser Boston Inc., Boston, MA, 1991.
[48] M.R. Stein, Whitehead groups of finite groups, Bull. Amer. Math. Soc. 84 (1978), 201–212.
[49] J. Stienstra, Operations in the higher K-theory of endomorphisms, Current trends in alge braic topology, Part 1 (London, Ont., 1981), CMS Conf. Proc., 2, Amer. Math. Soc., Providence, R.I., 1982, 59-–115.
[50] C. Walsh, Busemann points of Artin groups of dihedral type, Internat. J. Algebra Comput. 19 (2009), 891–910.
[51] C. Weibel, Mayer-Vietoris sequences and module structures on NK, Algebraic K-theory, Evanston 1980 (Proc. Conf., Northwestern Univ., Evanston, Ill., 1980), Lecture Notes in Math., 854, Springer, Berlin, 1981, 466—493.
[52] C. Weibel, Algebraic K-theory of rings of integers in local and global fields, Handbook of K-theory, volume 2, pp. 339–390, Springer, 2005.
[53] C. Weibel, NK0 and NK1 of the groups C4 and D4, Comment. Math. Helv. 84 (2009), 339-–349.
[54] C. Weibel, The K-book: an introduction to algebraic K-theory, Graduate Studies in Mathematics, 145 (2013), American Mathematical Society.