For any pair of orientable closed hyperbolic 3-manifolds, this paper shows that any isomorphism between the profinite completions of their fundamental groups witnesses a bijective correspondence between the Zariski dense $\text {PSL}(2,{\mathbb {Q}}^{{\textrm{ac}}})$-representations of their fundamental groups, up to conjugacy; moreover, corresponding pairs of representations have identical invariant trace fields and isomorphic invariant quaternion algebras. (Here, ${\mathbb {Q}}^{{\textrm{ac}}}$ denotes an algebraic closure of ${\mathbb {Q}}$.) Next, assuming the p-adic Borel regulator injectivity conjecture for number fields, this paper shows that uniform lattices in $\text {PSL}(2,{\mathbb {C}})$ with isomorphic profinite completions have identical invariant trace fields, isomorphic invariant quaternion algebras, identical covolume, and identical arithmeticity.
| [1] |
Agol, I.: The virtual Haken conjecture, with an appendix by Agol, Daniel Groves, and Jason Manning. Documenta Math. 18,1045–1087 (2013)
|
| [2] |
Aschenbrenner, M., Friedl, S., Wilton, H.: 3-Manifold Groups. EMS Series of Lectures in Mathematics, European Mathematical Society, Zürich (2015)
|
| [3] |
Besson G, Courtois G, Gallot S. Inégalités de Milnor–Wood géométriques. Comment. Math. Helv.. 2007, 82(4): 753-803
|
| [4] |
Borel A. Stable real cohomology of arithmetic groups. Ann. Sci. École Norm. Sup. (4). 1974, 7: 235-272
|
| [5] |
Borel, A., Wallach, N.: Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups. Math. Surveys and Monographs, vol. 67. American Mathematical Society, Providence, RI (2000)
|
| [6] |
Bridson, M.R., Reid, A.W.: Profinite rigidity, fibering and the figure-eight knot. In: What’s Next?—The Mathematical Legacy of William P. Thurston, Annals. Math. Stud., vol. 205, pp. 45–64. Princeton Univ. Press, Princeton, NJ (2020)
|
| [7] |
Bridson MR, McReynolds DB, Reid AW, Spitler R. Absolute profinite rigidity and hyperbolic geometry. Ann. Math. (2). 2020, 192(3): 679-719
|
| [8] |
Bridson MR, Reid AW, Wilton H. Profinite rigidity and surface bundles over the circle. Bull. Lond. Math. Soc.. 2017, 49(5): 831-841
|
| [9] |
Calegari F. The stable homology of congruence subgroups. Geom. Topol.. 2015, 19(6): 3149-3191
|
| [10] |
Culler M. Lifting representations to covering groups. Adv. Math.. 1986, 59(1): 64-70
|
| [11] |
Culler M, Shalen PB. Varieties of group representations and splittings of 3-manifolds. Ann. Math. (2). 1983, 117(1): 109-146
|
| [12] |
Derbez P, Liu Y, Sun H, Wang S. Volume of representations and mapping degree. Adv. Math.. 2019, 351: 570-613
|
| [13] |
Fathi, A., Laudenbach, F., Poénaru, V.: Thurston’s Work on Surfaces. Translated from the 1979 French original by Djun M. Kim and Dan Margalit. Math. Notes, vol. 48. Princeton University Press, Princeton, NJ (2012)
|
| [14] |
Funar L. Torus bundles not distinguished by TQFT invariants. Geom. Topol.. 2013, 17(4): 2289-2344
|
| [15] |
Hempel, J.: Some 3-manifold groups with the same finite quotients. arXiv:1409.3509v2 (2014)
|
| [16] |
Heusener, M., Porti, J.: The variety of characters in PSL$_2({\mathbb{C} })$. Bol. Soc. Mat. Mexicana (3) 10, 221–237 (2004)
|
| [17] |
Huber A, Kings G. A $p$-adic analogue of the Borel regulator and the Bloch–Kato exponential map. J. Inst. Math. Jussieu. 2011, 10(1): 149-190
|
| [18] |
Humphreys, J.E.: Linear Algebraic Groups. Graduate Texts in Mathematics, vol. 21. Springer-Verlag, New York-Heidelberg (1975)
|
| [19] |
Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory. Second printing, revised. Graduate Texts in Mathematics, vol. 9. Springer-Verlag, New York-Berlin (1978)
|
| [20] |
Jaikin-Zapirain A. Recognition of being fibered for compact 3-manifolds. Geom. Topol.. 2020, 24(1): 409-420
|
| [21] |
Jiang B, Guo J. Fixed points of surface diffeomorphisms. Pac. J. Math.. 1993, 160: 67-89
|
| [22] |
Kim S, Kim I. Volume invariant and maximal representations of discrete subgroups of Lie groups. Math. Z.. 2014, 276(3–4): 1189-1213
|
| [23] |
Koblitz, N.: $p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions. 2nd edn. Graduate Text in Mathematics, vol. 58, Springer-Verlag, New York (1984)
|
| [24] |
Liu Y. Virtual homological spectral radii for automorphisms of surfaces. J. Am. Math. Soc.. 2020, 33(4): 1167-1227
|
| [25] |
Liu Y. Finite-volume hyperbolic 3-manifolds are almost determined by their finite quotient groups. Invent. Math.. 2023, 231(2): 741-804
|
| [26] |
Liu Y, Sun H. Virtual 1-domination of 3-manifolds. Compositio Math.. 2018, 154(3): 621-639
|
| [27] |
Maclachlan, C., Reid, A.W.: The Arithmetic of Hyperbolic 3-Manifolds. Graduate Texts in Mathematics, vol. 219. Springer-Verlag, New York (2003)
|
| [28] |
Mahler K. Über transzendente $P$-adische Zahlen. Compositio Math.. 1935, 2: 259-275
|
| [29] |
Milnor, J.: Introduction to Algebraic K-Theory. Annals of Math. Studies, vol. 72. Princeton Univ. Press, Princeton (1971)
|
| [30] |
Nikolov N, Segal D. On finitely generated profinite groups. I. Strong completeness and uniform bounds. Ann. Math. (2). 2007, 165(1): 171-238
|
| [31] |
Ratcliffe J. Foundations of Hyperbolic Manifolds, Second Edition, Graduate Texts in Mathematics. 2006, New York, Springer149
|
| [32] |
Reid, A.W.: Profinite properties of discrete groups. In: Groups St Andrews 2013. London Math. Soc. Lecture Note Ser., vol. 422, pp. 73–104. Cambridge Univ. Press, Cambridge (2015)
|
| [33] |
Reid, A.W.: Profinite rigidity. In: Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. II. Invited lectures, pp. 1193–1216. World Sci. Publ., Hackensack, NJ (2018)
|
| [34] |
Ribes L, Zalesskii PA. Profinite Groups. 20102Berlin, Springer-Verlag
|
| [35] |
Rosenberg, J.: Algebraic K-Theory and Its Applications. Graduate Texts in Mathematics, vol. 147. Springer-Verlag, New York (1994)
|
| [36] |
Sah C-H. Homology of classical Lie groups made discrete. III. J. Pure Appl. Alg.. 1989, 56(3): 269-312
|
| [37] |
Stebe PF. Conjugacy separability of groups of integer matrices. Proc. Am. Math. Soc.. 1972, 32: 1-7
|
| [38] |
Symonds, P., Weigel, T.: Cohomology of p-adic analytic groups, In: New Horizons in Pro-p Groups. Progr. Math., vol. 184, pp. 349–410. Birkhäuser Boston, Boston, MA (2000)
|
| [39] |
Ueki J. Profinite rigidity for twisted Alexander polynomials. J. Reine Angew. Math.. 2021, 771: 171-192
|
| [40] |
Van der Kallen W. Homology stability for linear groups. Invent. Math.. 1980, 60(3): 269-295
|
| [41] |
Vinberg ÈB. The smallest field of definition of a subgroup of the group $\text{PSL} _2$. Mat. Sb.. 1993, 184(10): 53-66
|
| [42] |
Veldkamp GR. Ein Transzendenz-Satz für p-adische Zahlen. J. Lond. Math. Soc.. 1940, 15: 183-192
|
| [43] |
Wilkes G. Profinite rigidity for Seifert fibre spaces. Geom. Dedicata. 2017, 188: 141-163
|
| [44] |
Wilkes G. Profinite rigidity of graph manifolds and JSJ decompositions of 3-manifolds. J. Algebra. 2018, 502: 538-587
|
| [45] |
Wilkes G. Profinite rigidity of graph manifolds, II: Knots and mapping classes. Israel J. Math.. 2019, 233(1): 351-378
|
| [46] |
Wilton H, Zalesskii PA. Distinguishing geometries using finite quotients. Geom. Topol.. 2017, 21(1): 345-384
|
| [47] |
Wilton H, Zalesskii PA. Profinite detection of 3-manifold decompositions. Compos. Math.. 2019, 155(2): 246-259
|
Funding
National Outstanding Youth Science Fund Project of National Natural Science Foundation of China(11925101)
Key Technologies Research and Development Program(2020YFA0712800)
RIGHTS & PERMISSIONS
Peking University
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