Right-angled Artin groups and representation liftings
DOI:
https://doi.org/10.56764/hpu2.jos.2024.3.2.46-49Abstract
The lifting problems are interesting problems of number theory. There are many mathematicians who study lifting problems with different classes of groups. They prove the lifting problems with different classes of groups using various methods. Recently, right-angled Artin groups have attracted much attention in number theory. They have nice structure and properties. Currently, we study right-angled Artin groups with different problems related to them. One of those problems is that we want to prove the lifting problem is associated with this class of groups. We have obtained a result for this problem. In this paper, we will show that a mod Heisenberg representations of a right-angled Artin group can be lifted to a mod representation.
References
[1] C Khare, “Base change, lifting, and Serre’s conjecture,” J. Number Theory, vol. 63, no. 2, pp. 387–395, Apr. 1997, doi: 10.1006/jnth.1997.2093.
[2] C. B. Khare, M. Larsen, “Liftable groups, negligible cohomology and Heisenberg representations,” arXiv.org (Cornell University), Jan., 2020, doi: 10.4855/arxiv.2009.01301.
[3] C. De Clercq and M. Florence, “Lifting low-dimensional local systems,” Math. Z., vol. 300, no. 1, pp. 125–138, May. 2021, doi: 10.1007/s00209-021-02763-1.
[4] G. Böckle, “Lifting mod representations to characteristics ,” J. Number Theory, vol. 101, no. 2, pp. 310–337, Aug. 2003, doi: 10.1016/S0022-314X(03)00058-1.
[5] J. Mináˇc and N. D. Tân, “Triple Massey products over global fields,” Doc. Math., vol. 20, pp. 1467–1480, Jan. 2015, doi: 10.4171/dm/523.
[6] J. Mináˇc, M. Rogelstad, and N. D. Tân, “Dimensions of Zassenhaus filtration subquotients of some pro-p-groups,” Isr. J. Math., vol. 212, no. 2, pp. 825–855, May. 2016, doi: 10.1007/s11856-016-1310-0.
[7] J. Neukirch, A. Schmidt, and K. Wingberg, Eds. Cohomology of number fields (Grundlehren der mathematischen Wissenschaften). Heidelberg, Germany: Springer, 2008, doi: 10.1007/978-3-540-37889-1.
[8] R. Ramakrishna, “Lifting Galois representations,” Invent. Math., vol. 138, no. 3, pp. 537–562, Dec. 1999, doi: 10.1007/s002220050352.
[9] R. D. Wade, “The lower central series of a right-angled Artin group,” Enseign. Math., vol. 61, no. 3, pp. 343–371, Aug. 2016, doi: 10.4171/lem/61-3/4-4.
[10] C. Jensen and J. Meier, “The cohomology of right-angled artin groups with group ring coefficients,” Bull. Lond. Math. Soc., vol. 37, no. 5, pp. 711–718, Oct. 2005, doi: 10.1112/S0024609305004571.
[11] A. Baudisch, “Subgroups of semifree groups,” Acta Math. Acad. Sci. Hungar., vol. 38, no. 1–4, pp. 19–28, Mar. 1981, doi: 10.1007/BF01917515.
[12] I. Efrat and E. Matzri, “Vanishing of massey products and Brauer groups,” Can. Math. Bull., vol. 58, no. 4, pp. 730–740, Dec. 2015, doi: 10.4153/cmb-2015-026-5.
[13] T. Hsu and D. T. Wise, “Separating quasiconvex subgroups of right-angled Artin groups,” Math. Z., vol. 240, no. 3, pp. 521–548, Jul. 2002, doi: 10.1007/s002090100329.
[14] R. Fox and L. Neuwirth, “The Braid groups,” Math. Scand., vol. 10, p. 119, Jun. 1962, doi: 10.7146/math.scand.a-10518.
[15] D. J. Allcock, “Braid pictures for Artin groups,” Trans. Am. Math. Soc., vol. 354, no. 9, pp. 3455–3474, Apr. 2002, doi: 10.1090/S0002-9947-02-02944-6.
[16] F. Garside, “The braid group and other groups,” Q. J. Math., vol. 20, no. 1, pp. 235–254, Jan. 1969, doi: 10.1093/qmath/20.1.235.
[17] T. Koberda, “Right-angled Artin groups and a generalized isomorphism problem for finitely generated subgroups of mapping class groups,” Geom. Funct. Anal., vol. 22, no. 6, pp. 1541–1590, Sep. 2012, doi: 10.1007/s00039-012-0198-z.
[18] S. Papadima and A. I. Suciu, “Algebraic invariants for right-angled Artin groups,” Math. Ann., vol. 334, no. 3, pp. 533–555, Dec. 2005, doi: 10.1007/s00208-005-0704-9.
[19] R. J. Wilson, “Introduction to graph theory”, in Oxford university press eBooks, Edinburgh, England: Addison Wesley Longman, 1997, ch. 1, pp. 1–12, doi: 10.1093/oso/9780198514978.003.0001.
[20] H. Koch, Ed. Galois theory of p-extensions (Springer Monographs in Mathematics). Heidelberg, Germany: Springer, 2002, doi: 10.1007/978-3-662-04967-9.
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Copyright (c) 2024 Thi-Tra Nguyen, Kim-Thuy Dinh Thi, Huu-Linh Nguyen
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