Alexander invariants and cohomology jump loci in group extensions. (English) Zbl 07917522
Summary: We study the integral, rational, and modular Alexander invariants, as well as the cohomology jump loci of groups arising as extensions with trivial algebraic monodromy. Our focus is on extensions of the form \(1\to K \to G \to Q \to 1\), where \(Q\) is an Abelian group acting trivially on \(H_1 (K;\mathbb{Z})\), with suitable modifications in the rational and mod-\(p\) settings. We find a tight relationship between the Alexander invariants, the characteristic varieties, and the resonance varieties of the groups \(K\) and \(G\). This leads to an inequality between the respective Chen ranks, which becomes an equality in degrees greater than1for split extensions.
MSC:
| 55N25 | Homology with local coefficients, equivariant cohomology |
| 14M12 | Determinantal varieties |
| 16W70 | Filtered associative rings; filtrational and graded techniques |
| 17B70 | Graded Lie (super)algebras |
| 20F14 | Derived series, central series, and generalizations for groups |
| 20F40 | Associated Lie structures for groups |
| 20J05 | Homological methods in group theory |
| 57M05 | Fundamental group, presentations, free differential calculus |
| 57M07 | Topological methods in group theory |
References:
| [1] | M. APRODU, G. FARKAS, S. PAPADIMA, C. RAICU and J. WEYMAN, Topological in-variants of groups and Koszul modules, Duke Math. J. 171 (2022), 2013-2046. · Zbl 1514.57028 |
| [2] | M. APRODU, G. FARKAS, S. PAPADIMA, C. RAICU and J. WEYMAN, Koszul modules and Green’s Conjecture, Invent. Math. 218 (2019), 657-720. · Zbl 1430.14074 |
| [3] | M. APRODU, G. FARKAS, C. RAICU and A. SUCIU, Reduced resonance schemes and Chen ranks, J. Reine Angew. Math. (to appear), available at ArXiv:2303.07855v2. |
| [4] | M. APRODU, G. FARKAS, C. RAICU and A. SUCIU, The effective Chen ranks conjecture, preprint (2024). |
| [5] | H. BASS and A. LUBOTZKY, Linear-central filtrations on groups, In: “The Mathemat-ical Legacy of Wilhelm Magnus: Groups, Geometry and Special Functions” (Brooklyn, NY, 1992), W. Abikoff, J. S. Birman and K. Kuikern (eds.), Contemporary Mathematics, Vol. 169, American Mathematical Society, Providence, RI, 1994, 45-98. · Zbl 0817.20038 |
| [6] | P. BELLINGERI and S. GERVAIS, On p-almost direct products and residual properties of pure braid groups of nonorientable surfaces, Algebr. Geom. Topol. 16 (2016), 547-568. · Zbl 1376.20035 |
| [7] | P. BELLINGERI, S. GERVAIS and J. GUASCHI, Exact sequences, lower central series and representations of surface braid groups, preprint ArXiv:1106.4982v1 (2011). |
| [8] | M. BESTVINA and N. BRADY, Morse theory and finiteness properties of groups, Invent. Math. 129 (1997), 445-470. · Zbl 0888.20021 |
| [9] | K. S. BROWN, “Cohomology of Groups”, Corrected reprint of the 1982 original, Graduate Texts in Mathematics, Vol. 87, Springer-Verlag, New York, 1994. · Zbl 0584.20036 |
| [10] | K.-T. CHEN, Integration in free groups, Ann. of Math. (2) 54 (1951), 147-162. · Zbl 0045.30102 |
| [11] | K.-T. CHEN, Extension of C 1 function algebra by integrals and Malcev completion of ⇡ 1 , Advances in Math. 23 (1977),181-210. · Zbl 0345.58003 |
| [12] | T. CHURCH, M. ERSHOV and A. PUTMAN, On finite generation of the Johnson filtrations, J. Eur. Math. Soc. (JEMS) 24 (2022), 2875-2914. · Zbl 07523092 |
| [13] | T. COCHRAN, Noncommutative knot theory, Algebr. Geom. Topol. 4 (2004), 347-398. · Zbl 1063.57011 |
| [14] | T. COCHRAN and S. HARVEY, Homology and derived series of groups, Geom. Topol. 9 (2005), 2159-2191. · Zbl 1138.20043 |
| [15] | T. COCHRAN and S. HARVEY, Homology and derived p-series of groups, J. Lond. Math. Soc. 78 (2008), 677-692. · Zbl 1159.57004 |
| [16] | T. COCHRAN and S. HARVEY, Homological stability of series of groups, Pacific J. Math. 246 (2010), 31-47. · Zbl 1229.20050 |
| [17] | D. C. COHEN and H. K. SCHENCK, Chen ranks and resonance, Adv. Math. 285 (2015), 1-27. · Zbl 1360.20021 |
| [18] | D. C. COHEN and A. I. SUCIU, Alexander invariants of complex hyperplane arrange-ments, Trans. Amer. Math. Soc. 351 (1999), 4043-4067. · Zbl 0945.20024 |
| [19] | J. COOPER, Two mod-p Johnson filtrations, J. Topol. Anal. 7 (2015), 309-343. · Zbl 1316.57001 |
| [20] | R. H. CROWELL, Torsion in link modules, J. Math. Mech. 14 (1965), 289-298. · Zbl 0134.43102 |
| [21] | G. DENHAM, A. I. SUCIU and S. YUZVINSKY, Abelian duality and propagation of reso-nance, Selecta Math. 23 (2017), 2331-2367. · Zbl 1381.55005 |
| [22] | A. DIMCA, R. HAIN and S. PAPADIMA, The abelianization of the Johnson kernel, J. Eur. Math. Soc. (JEMS) 16 (2014) 805-822. · Zbl 1344.57001 |
| [23] | A. DIMCA and S. PAPADIMA, Finite Galois covers, cohomology jump loci, formality prop-erties, and multinets, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 10 (2011), 253-268. · Zbl 1239.32023 |
| [24] | A. DIMCA and S. PAPADIMA, Arithmetic group symmetry and finiteness properties of Torelli groups, Ann. of Math. (2) 177 (2013), 395-423. · Zbl 1271.57053 |
| [25] | A. DIMCA, S. PAPADIMA and A. I. SUCIU, Formality, Alexander invariants, and a ques-tion of Serre, preprint ArXiv:math.AT/0512480v3 (2005). |
| [26] | A. DIMCA, S. PAPADIMA and A. I. SUCIU, Alexander polynomials: Essential variables and multiplicities, Int. Math. Res. Not. IMRN (2008), Art. ID rnm119, 36 pp. · Zbl 1156.32018 |
| [27] | A. DIMCA, S. PAPADIMA and A. I. SUCIU, Topology and geometry of cohomology jump loci, Duke Math. J. 148 (2009), 405-457. · Zbl 1222.14035 |
| [28] | W. G. DWYER and D. FRIED, Homology of free Abelian covers. I, Bull. London Math. Soc. 19 (1987), 350-352. · Zbl 0625.57001 |
| [29] | D. EISENBUD, “Commutative Algebra, with a View Towards Algebraic Geometry”, Grad-uate Texts in Mathematics, Vol. 150, Springer-Verlag, New York, 1995. · Zbl 0819.13001 |
| [30] | D. EISENBUD, “The Geometry of Syzygies, a Second Course in Commutative Algebra and Algebraic Geometry”, Graduate Texts in Mathematics, Vol. 229, Springer-Verlag, New York, 2005. · Zbl 1066.14001 |
| [31] | D. EISENBUD and W. NEUMANN, “Three-Dimensional Link Theory and Invariants of Plane Curve Singularities”, Annals of Mathematics Studies, Vol. 110, Princeton Univer-sity Press, Princeton, NJ, 1985. · Zbl 0628.57002 |
| [32] | M. FALK and R. RANDELL, The lower central series of a fiber-type arrangement, Invent. Math. 82 (1985), 77-88. · Zbl 0574.55010 |
| [33] | J. GUASCHI and C. M. PEREIRO, Lower central and derived series of semi-direct prod-ucts, and applications to surface braid groups, J. Pure Appl. Algebra 224 (2020), Paper No. 106309, 39 pp. · Zbl 1452.20033 |
| [34] | R. HAIN, Relative weight filtrations on completions of mapping class groups, In: “Groups of Diffeomorphisms”, R. Penner, D. Kotschick, T. Tsuboi, N. Kawazumi, T. Kitano and Y. Mitsumatsu (eds.), Advanced Studies in Pure Mathematics, Vol. 52, Mathematical Soci-ety of Japan, Tokyo, 2008, 309-368. · Zbl 1176.57002 |
| [35] | P. HALL, A contribution to the theory of groups of prime-power order, Proc. Lond. Math. Soc., II. Ser. 36 (1933), 29-95. · Zbl 0007.29102 |
| [36] | S. L. HARVEY, Higher-order polynomial invariants of 3-manifolds giving lower bounds for the Thurston norm, Topology 44 (2005), 895-945. · Zbl 1080.57019 |
| [37] | W. HERFORT, K. H. HOFMANN and F. G. RUSSO, “Periodic Locally Compact Groups”, De Gruyter Studies Mathematics, Vol. 71, De Gruyter, Berlin, 2019. · Zbl 1423.22001 |
| [38] | P. J. HILTON and U. STAMMBACH, “A Course in Homological Algebra”, Second ed., Graduate Texts in Mathematics, Vol. 4, Springer-Verlag, New York, 1997. · Zbl 0863.18001 |
| [39] | E. HIRONAKA, Alexander stratifications of character varieties, Ann. Inst. Fourier (Greno-ble) 47 (1997), 555-583. · Zbl 0870.57003 |
| [40] | G. KARPILOVSKY, “The Schur Multiplier”, London Mathematical Society Monograph (N.S.), Vol. 2, Clarendon Press, Oxford University Press, New York, 1987. · Zbl 0619.20001 |
| [41] | M. LACKENBY, New lower bounds on subgroup growth and homology growth, Proc. Lond. Math. Soc. (3) 98 (2009), 271-297. · Zbl 1175.20025 |
| [42] | M. LACKENBY, Detecting large groups, J. Algebra 324 (2010), 2636-2657. · Zbl 1231.20026 |
| [43] | M. LAZARD, Sur les groupes nilpotents et les anneaux de Lie, Ann. Sci. École Norm. Sup. (3) 71 (1954), 101-190. · Zbl 0055.25103 |
| [44] | C. R. LEEDHAM-GREEN and S. MCKAY, “The Structure of Groups of Prime Power Or-der”, London Mathematical Society Monograph (N.S.), Vol. 27. Oxford University Press, Oxford, 2002. · Zbl 1008.20001 |
| [45] | J. C. LENNOX and D. J. S. ROBINSON, “The Theory of Infinite Soluble Groups”, The Clarendon Press, Oxford University Press, Oxford, 2004. · Zbl 1059.20001 |
| [46] | A. LIBGOBER, On the homology of finite Abelian coverings, Topology. Appl. 43 (1992), 157-166. · Zbl 0770.14004 |
| [47] | W. MAGNUS, A. KARRASS and D. SOLITAR, “Combinatorial Group Theory”, second re-vised edition, Dover, New York, 1976. · Zbl 0362.20023 |
| [48] | M. MARKL and S. PAPADIMA, Homotopy Lie algebras and fundamental groups via defor-mation theory, Ann. Inst. Fourier (Grenoble) 42 (1992), 905-935. · Zbl 0760.55010 |
| [49] | W. S. MASSEY, Completion of link modules, Duke Math. J. 47 (1980), 399-420. · Zbl 0464.57001 |
| [50] | G. MASSUYEAU, Finite-type invariants of 3-manifolds and the dimension subgroup prob-lem, J. Lond. Math. Soc. (2) 75 (2007), 791-811. · Zbl 1129.57017 |
| [51] | D. MATEI and A. I. SUCIU, Homotopy types of complements of 2-arrangements in R 4 , Topology 39 (2000), 61-88. · Zbl 0940.55010 |
| [52] | D. MATEI and A. I. SUCIU, Cohomology rings and nilpotent quotients of real and complex arrangements, In: “Arrangements -Tokyo 1998”, M. Falk and H. Terao (eds.), Advanced Studies Pure Mathematics, Vol. 27, Mathematical Society of Japan, Tokyo, 2000, 185-215. · Zbl 0974.32020 |
| [53] | D. MATEI and A. I. SUCIU, Hall invariants, homology of subgroups, and characteristic varieties, Int. Math. Res. Not. (2002), 465-503. · Zbl 1061.20040 |
| [54] | S. PAPADIMA and A. I. SUCIU, Chen Lie algebras, Int. Math. Res. Not. (2004), 1057-1086. · Zbl 1076.17007 |
| [55] | S. PAPADIMA and A. I. SUCIU, Algebraic invariants for right-angled Artin groups, Math. Ann. 334 (2006), 533-555. · Zbl 1165.20032 |
| [56] | S. PAPADIMA and A. I. SUCIU, Algebraic invariants for Bestvina-Brady groups, J. Lond. Math. Soc. (2) 76 (2007), 273-292. · Zbl 1176.20038 |
| [57] | S. PAPADIMA and A. I. SUCIU, Toric complexes and Artin kernels, Adv. Math. 220 (2009), 441-477. · Zbl 1208.57002 |
| [58] | S. PAPADIMA and A. SUCIU, Geometric and algebraic aspects of 1-formality, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 52 (2009), 355-375. · Zbl 1199.55010 |
| [59] | S. PAPADIMA and A. I. SUCIU, Bieri-Neumann-Strebel-Renz invariants and homology jumping loci, Proc. London Math. Soc. 100 (2010), 795-834. · Zbl 1273.55003 |
| [60] | S. PAPADIMA and A. I. SUCIU, Homological finiteness in the Johnson filtration of the au-tomorphism group of a free group, J. Topol. 5 (2012), 909-944. · Zbl 1268.20037 |
| [61] | S. PAPADIMA and A. I. SUCIU, Jump loci in the equivariant spectral sequence, Math. Res. Lett. 21 (2014), 863-883. · Zbl 1314.55002 |
| [62] | S. PAPADIMA and A. I. SUCIU, Vanishing resonance and representations of Lie algebras, J. Reine Angew. Math. 706 (2015), 83-101. · Zbl 1372.17013 |
| [63] | L. PARIS, Residual p properties of mapping class groups and surface groups, Trans. Amer. Math. Soc. 361 (2009), 2487-2507. · Zbl 1248.20042 |
| [64] | D. S. PASSMAN, “The Algebraic Structure of Group Rings”, Pure and Applied Mathemat-ics, Wiley-Interscience, New York-London-Sydney, 1977. · Zbl 0368.16003 |
| [65] | D. G. QUILLEN, On the associated graded ring of a group ring, J. Algebra 10 (1968), 411-418. · Zbl 0192.35803 |
| [66] | D. QUILLEN, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205-295. · Zbl 0191.53702 |
| [67] | P. B. SHALEN and P. WAGREICH, Growth rates, Z p -homology, and volumes of hyperbolic 3-manifolds, Trans. Amer. Math. Soc. 331 (1992), 895-917. · Zbl 0768.57001 |
| [68] | J. STALLINGS, Homology and central series of groups, J. Algebra 2 (1965), 170-181. · Zbl 0135.05201 |
| [69] | J. R. STALLINGS, Surfaces in three-manifolds and nonsingular equations in groups, Math. Z. 184 (1983), 1-17. · Zbl 0496.57006 |
| [70] | A. I. SUCIU, Fundamental groups of line arrangements: enumerative aspects, In: “Ad-vances in Algebraic Geometry Motivated by Physics” (Lowell, MA, 2000), Contemporary Mathematics, Vol. 276, American Mathematical Society, Providence, RI, 2001, 43-79. · Zbl 0998.14012 |
| [71] | A. I. SUCIU, Characteristic varieties and Betti numbers of free Abelian covers, Int. Math. Res. Notices IMRN 2014 (2014), 1063-1124. · Zbl 1352.57004 |
| [72] | A. I. SUCIU, Hyperplane arrangements and Milnor fibrations, Ann. Fac. Sci. Toulouse Math. (6) 23 (2014), 417-481. · Zbl 1300.32028 |
| [73] | A. I. SUCIU, On the topology of Milnor fibrations of hyperplane arrangements, Rev. Roumaine Math. Pures Appl. 62 (2017), 191-215. · Zbl 1389.32030 |
| [74] | A. I. SUCIU, Poincaré duality and resonance varieties, Proc. Roy. Soc. Edinburgh Sect. A 150 (2020), 3001-3027. · Zbl 1461.55015 |
| [75] | A. I. SUCIU, Lower central series and split extensions, preprint ArXiv:2105.14129v2. |
| [76] | A. I. SUCIU, Milnor fibrations of arrangements with trivial algebraic monodromy, Rev. Roumaine Math. Pures Appl. (to appear), available at ArXiv:2402.03619v2. |
| [77] | A. I. SUCIU and H. WANG, Pure virtual braids, resonance, and formality, Math. Z. 286 (2017), 1495-1524. · Zbl 1423.20034 |
| [78] | A. I. SUCIU and H. WANG, Cup products, lower central series, and holonomy Lie alge-bras, J. Pure Appl. Algebra 223 (2019), 3359-3385. · Zbl 1515.20177 |
| [79] | A. I. SUCIU and H. WANG, Formality properties of finitely generated groups and Lie al-gebras, Forum Math. 31 (2019), 867-905. · Zbl 1454.20075 |
| [80] | A. I. SUCIU and H. WANG, Chen ranks and resonance varieties of the upper McCool groups, Adv. in Appl. Math. 110 (2019), 197-234. · Zbl 1480.20078 |
| [81] | A. I. SUCIU and H. WANG, Taylor expansions of groups and filtered-formality, Eur. J. Math. 6 (2020), 1073-1096. · Zbl 1530.20110 |
| [82] | A. I. SUCIU, Y. YANG and G. ZHAO, Homological finiteness of Abelian covers, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 14 (2015), 101-153. Department of Mathematics Northeastern University Boston, MA 02115, USA a.suciu@northeastern.edu · Zbl 1349.55003 |
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