From the Poincaré Theorem to generators of the unit group of integral group rings of finite groups

Jespers, E.; Juriaans, S.; Kiefer, A.; de A. e Silva, A.; Souza Filho, A.

doi:10.1090/S0025-5718-2014-02865-2

From the Poincaré Theorem to generators of the unit group of integral group rings of finite groups
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by E. Jespers, S. O. Juriaans, A. Kiefer, A. de A. e Silva and A. C. Souza Filho PDF

Math. Comp. 84 (2015), 1489-1520 Request permission

Abstract:

We give an algorithm to determine finitely many generators for a subgroup of finite index in the unit group of an integral group ring $\mathbb {Z} G$ of a finite nilpotent group $G$, this provided the rational group algebra $\mathbb {Q} G$ does not have simple components that are division classical quaternion algebras or two-by-two matrices over a classical quaternion algebra with center $\mathbb {Q}$. The main difficulty is to deal with orders in quaternion algebras over the rationals or a quadratic imaginary extension of the rationals. In order to deal with these we give a finite and easy implementable algorithm to compute a polyhedron containing a fundamental domain in the hyperbolic three space $\mathbb {H}^3$ (respectively, hyperbolic two space $\mathbb {H}^2$) for a discrete subgroup of $\mathrm {PSL}_2(\mathbb {C})$ (respectively, $\mathrm {PSL}_2(\mathbb {R})$) of finite covolume. Our results on group rings are a continuation of earlier work of Ritter and Sehgal, Jespers and Leal.

References

S. A. Amitsur, Finite subgroups of division rings, Trans. Amer. Math. Soc. 80 (1955), 361–386. MR 74393, DOI 10.1090/S0002-9947-1955-0074393-9
Anthony Bak and Ulf Rehmann, The congruence subgroup and metaplectic problems for $\textrm {SL}_{n\geq 2}$ of division algebras, J. Algebra 78 (1982), no. 2, 475–547. MR 680373, DOI 10.1016/0021-8693(82)90094-1
Behnam Banieqbal, Classification of finite subgroups of $2\times 2$ matrices over a division algebra of characteristic zero, J. Algebra 119 (1988), no. 2, 449–512. MR 971143, DOI 10.1016/0021-8693(88)90069-5
Hyman Bass, The Dirichlet unit theorem, induced characters, and Whitehead groups of finite groups, Topology 4 (1965), 391–410. MR 193120, DOI 10.1016/0040-9383(66)90036-X
H. Bass, J. Milnor, and J.-P. Serre, Solution of the congruence subgroup problem for $\textrm {SL}_{n}\,(n\geq 3)$ and $\textrm {Sp}_{2n}\,(n\geq 2)$, Inst. Hautes Études Sci. Publ. Math. 33 (1967), 59–137. MR 244257
Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1995. Corrected reprint of the 1983 original. MR 1393195
Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486, DOI 10.1007/978-3-662-12494-9
Capi Corrales, Eric Jespers, Guilherme Leal, and Angel del Río, Presentations of the unit group of an order in a non-split quaternion algebra, Adv. Math. 186 (2004), no. 2, 498–524. MR 2073916, DOI 10.1016/j.aim.2003.07.015
Ann Dooms, Eric Jespers, and Alexander Konovalov, From Farey symbols to generators for subgroups of finite index in integral group rings of finite groups, J. K-Theory 6 (2010), no. 2, 263–283. MR 2735087, DOI 10.1017/is009012013jkt079
J. Elstrodt, F. Grunewald, and J. Mennicke, Groups acting on hyperbolic space, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. Harmonic analysis and number theory. MR 1483315, DOI 10.1007/978-3-662-03626-6
Dieter Flöge, Zur Struktur der $\textrm {PSL}_{2}$ über einigen imaginär-quadratischen Zahlringen, Math. Z. 183 (1983), no. 2, 255–279 (German). MR 704107, DOI 10.1007/BF01214824
Antonio Giambruno and Sudarshan K. Sehgal, Generators of large subgroups of units of integral group rings of nilpotent groups, J. Algebra 174 (1995), no. 1, 150–156. MR 1332864, DOI 10.1006/jabr.1995.1121
M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI 10.1007/978-1-4613-9586-7_{3}
E. Jespers, S. O. Juriaans, A. Kiefer, A. de A. e Silva, and A. C. Souza Filho, Poincaré bisectors in hyperbolic spaces, preprint.
E. Jespers and G. Leal, Degree $1$ and $2$ representations of nilpotent groups and applications to units of group rings, Manuscripta Math. 86 (1995), no. 4, 479–498. MR 1324684, DOI 10.1007/BF02568007
Eric Jespers and Guilherme Leal, Generators of large subgroups of the unit group of integral group rings, Manuscripta Math. 78 (1993), no. 3, 303–315. MR 1206159, DOI 10.1007/BF02599315
E. Jespers, M. M. Parmenter, and S. K. Sehgal, Central units of integral group rings of nilpotent groups, Proc. Amer. Math. Soc. 124 (1996), no. 4, 1007–1012. MR 1328353, DOI 10.1090/S0002-9939-96-03398-9
Eric Jespers, Gabriela Olteanu, and Ángel del Río, Rational group algebras of finite groups: from idempotents to units of integral group rings, Algebr. Represent. Theory 15 (2012), no. 2, 359–377. MR 2892512, DOI 10.1007/s10468-010-9244-4
Eric Jespers, Antonio Pita, Ángel del Río, Manuel Ruiz, and Pavel Zalesskii, Groups of units of integral group rings commensurable with direct products of free-by-free groups, Adv. Math. 212 (2007), no. 2, 692–722. MR 2329317, DOI 10.1016/j.aim.2006.11.005
Stefan Johansson, On fundamental domains of arithmetic Fuchsian groups, Math. Comp. 69 (2000), no. 229, 339–349. MR 1665958, DOI 10.1090/S0025-5718-99-01167-9
S. O. Juriaans, I. B. S. Passi, and Dipendra Prasad, Hyperbolic unit groups, Proc. Amer. Math. Soc. 133 (2005), no. 2, 415–423. MR 2093062, DOI 10.1090/S0002-9939-04-07578-1
S. O. Juriaans, I. B. S. Passi, and A. C. Souza Filho, Hyperbolic unit groups and quaternion algebras, Proc. Indian Acad. Sci. Math. Sci. 119 (2009), no. 1, 9–22. MR 2508485, DOI 10.1007/s12044-009-0002-7
S. O. Juriaans and A. C. Souza Filho, Free groups in quaternion algebras, J. Algebra 379 (2013), 314–321. MR 3019259, DOI 10.1016/j.jalgebra.2012.12.025
Svetlana Katok, Reduction theory for Fuchsian groups, Math. Ann. 273 (1986), no. 3, 461–470. MR 824433, DOI 10.1007/BF01450733
E. Kleinert, Two theorems on units of orders, Abh. Math. Sem. Univ. Hamburg 70 (2000), 355–358. MR 1809557, DOI 10.1007/BF02940925
Ernst Kleinert, Units of classical orders: a survey, Enseign. Math. (2) 40 (1994), no. 3-4, 205–248. MR 1309127
Ernst Kleinert, Units in skew fields, Progress in Mathematics, vol. 186, Birkhäuser Verlag, Basel, 2000. MR 1753508, DOI 10.1007/978-3-0348-8409-9
Bernhard Liehl, On the group $\textrm {SL}_{2}$ over orders of arithmetic type, J. Reine Angew. Math. 323 (1981), 153–171. MR 611449, DOI 10.1515/crll.1981.323.153
Colin Maclachlan and Alan W. Reid, The arithmetic of hyperbolic 3-manifolds, Graduate Texts in Mathematics, vol. 219, Springer-Verlag, New York, 2003. MR 1937957, DOI 10.1007/978-1-4757-6720-9
Gabriela Olteanu and Ángel del Río, Group algebras of Kleinian type and groups of units, J. Algebra 318 (2007), no. 2, 856–870. MR 2371975, DOI 10.1016/j.jalgebra.2007.03.026
Donald Passman, Permutation groups, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0237627
César Polcino Milies and Sudarshan K. Sehgal, An introduction to group rings, Algebra and Applications, vol. 1, Kluwer Academic Publishers, Dordrecht, 2002. MR 1896125, DOI 10.1007/978-94-010-0405-3
John G. Ratcliffe, Foundations of hyperbolic manifolds, 2nd ed., Graduate Texts in Mathematics, vol. 149, Springer, New York, 2006. MR 2249478
U. Rehmann, A survey of the congruence subgroup problem, Algebraic $K$-theory, Part I (Oberwolfach, 1980) Lecture Notes in Math., vol. 966, Springer, Berlin-New York, 1982, pp. 197–207. MR 689376
Robert Riley, Applications of a computer implementation of Poincaré’s theorem on fundamental polyhedra, Math. Comp. 40 (1983), no. 162, 607–632. MR 689477, DOI 10.1090/S0025-5718-1983-0689477-2
Jürgen Ritter and Sudarshan K. Sehgal, Construction of units in group rings of monomial and symmetric groups, J. Algebra 142 (1991), no. 2, 511–526. MR 1127078, DOI 10.1016/0021-8693(91)90322-Y
Jürgen Ritter and Sudarshan K. Sehgal, Construction of units in integral group rings of finite nilpotent groups, Trans. Amer. Math. Soc. 324 (1991), no. 2, 603–621. MR 987166, DOI 10.1090/S0002-9947-1991-0987166-9
Jürgen Ritter and Sudarshan K. Sehgal, Units of group rings of solvable and Frobenius groups over large rings of cyclotomic integers, J. Algebra 158 (1993), no. 1, 116–129. MR 1223670, DOI 10.1006/jabr.1993.1126
S. K. Sehgal, Units in integral group rings, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 69, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. With an appendix by Al Weiss. MR 1242557
S. K. Sehgal, Group rings, Handbook of algebra, Vol. 3, Handb. Algebr., vol. 3, Elsevier/North-Holland, Amsterdam, 2003, pp. 455–541. MR 2035104, DOI 10.1016/S1570-7954(03)80069-4
M. Shirvani and B. A. F. Wehrfritz, Skew linear groups, London Mathematical Society Lecture Note Series, vol. 118, Cambridge University Press, Cambridge, 1986. MR 883801
Richard G. Swan, Generators and relations for certain special linear groups, Advances in Math. 6 (1971), 1–77 (1971). MR 284516, DOI 10.1016/0001-8708(71)90027-2
L. N. Vaseršteĭn, Structure of the classical arithmetic groups of rank greater than $1$, Mat. Sb. (N.S.) 91(133) (1973), 445–470, 472 (Russian). MR 0349864
T. N. Venkataramana, On systems of generators of arithmetic subgroups of higher rank groups, Pacific J. Math. 166 (1994), no. 1, 193–212. MR 1306038