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An \(L_2\)-quotient algorithm for finitely presented groups on arbitrarily many generators. (English) Zbl 1315.20034

Summary: We generalize the Plesken-Fabiańska \(L_2\)-quotient algorithm [W. Plesken and A. Fabiańska, J. Algebra 322, No. 3, 914-935 (2009; Zbl 1253.20033)] for finitely presented groups on two or three generators to allow an arbitrary number of generators. The main difficulty lies in a constructive description of the invariant ring of \(\mathrm{GL}(2,K)\) on \(m\) copies of \(\mathrm{SL}(2,K)\) by simultaneous conjugation. By giving this description, we generalize and simplify some of the known results in invariant theory. An implementation of the algorithm is available in the computer algebra system Magma.

MSC:

20F05 Generators, relations, and presentations of groups
13A50 Actions of groups on commutative rings; invariant theory
20-04 Software, source code, etc. for problems pertaining to group theory
68W30 Symbolic computation and algebraic computation

Citations:

Zbl 1253.20033

Software:

Magma; QuillenSuslin
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Full Text: DOI arXiv

References:

[1] Bosma, Wieb; Cannon, John; Playoust, Catherine, The Magma algebra system. I. The user language, Computational Algebra and Number Theory, London, 1993. Computational Algebra and Number Theory, London, 1993, J. Symbolic Comput., 24, 3-4, 235-265 (1997) · Zbl 0898.68039
[2] Brumfiel, G. W.; Hilden, H. M., \(SL(2)\) Representations of Finitely Presented Groups, Contemp. Math., vol. 187 (1995), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0838.20006
[3] Marston Conder; Havas, George; Newman, M. F., On one-relator quotients of the modular group, (Groups St Andrews 2009 in Bath, vol. 1. Groups St Andrews 2009 in Bath, vol. 1, London Math. Soc. Lecture Note Ser., vol. 387 (2011), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 183-197 · Zbl 1243.20062
[5] Conder, Marston, Group actions on the cubic tree, J. Algebraic Combin., 1, 3, 209-218 (1992) · Zbl 0779.05023
[6] Coxeter, H. S.M., The abstract groups \(G^{m, n, p}\), Trans. Amer. Math. Soc., 45, 1, 73-150 (1939) · JFM 65.0072.03
[7] Donkin, Stephen, Invariants of several matrices, Invent. Math., 110, 2, 389-401 (1992) · Zbl 0826.20036
[8] Drensky, Vesselin, Defining relations for the algebra of invariants of \(2 \times 2\) matrices, Algebr. Represent. Theory, 6, 2, 193-214 (2003) · Zbl 1026.16009
[9] Edjvet, M.; Juhász, A., The groups \(G^{m, n, p}\), J. Algebra, 319, 1, 248-266 (2008) · Zbl 1182.20035
[10] Fabiańska, Anna, Algorithmic analysis of presentations of groups and modules (2009), RWTH Aachen University, PhD thesis
[11] Fricke, Robert; Klein, Felix, Vorlesungen über die Theorie der automorphen Funktionen. Band 1: Die gruppentheoretischen Grundlagen. Band II: Die funktionentheoretischen Ausführungen und die Andwendungen, Bibliotheca Mathematica Teubneriana, Bände 3, vol. 4 (1965), Johnson Reprint Corp.: Johnson Reprint Corp. New York
[12] Glasby, S. P.; Leedham-Green, C. R.; O’Brien, E. A., Writing projective representations over subfields, J. Algebra, 295, 1, 51-61 (2006) · Zbl 1103.20009
[13] Havas, George; Holt, Derek F., On Coxeter’s families of group presentations, J. Algebra, 324, 5, 1076-1082 (2010) · Zbl 1204.20039
[14] Horowitz, Robert D., Characters of free groups represented in the two-dimensional special linear group, Comm. Pure Appl. Math., 25, 635-649 (1972) · Zbl 1184.20009
[15] Holt, Derek F.; Rees, Sarah, Testing modules for irreducibility, J. Aust. Math. Soc. A, 57, 1, 1-16 (1994) · Zbl 0833.20021
[16] Jambor, Sebastian, Computing minimal associated primes in polynomial rings over the integers, J. Symbolic Comput., 46, 10, 1098-1104 (2011) · Zbl 1230.13025
[17] Jambor, Sebastian, Determining Aschbacher classes using characters (2014), preprint · Zbl 1334.20005
[18] Macbeath, A. M., Generators of the linear fractional groups, (Number Theory. Number Theory, Houston, TX, 1967. Number Theory. Number Theory, Houston, TX, 1967, Proc. Sympos. Pure Math., vol. XII (1969), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 14-32
[19] Magnus, Wilhelm, Rings of Fricke characters and automorphism groups of free groups, Math. Z., 170, 1, 91-103 (1980) · Zbl 0433.20033
[20] Plesken, Wilhelm; Fabiańska, Anna, An \(L_2\)-quotient algorithm for finitely presented groups, J. Algebra, 322, 3, 914-935 (2009) · Zbl 1253.20033
[21] Procesi, C., The invariant theory of \(n \times n\) matrices, Adv. Math., 19, 3, 306-381 (1976) · Zbl 0331.15021
[22] Procesi, C., Computing with \(2 \times 2\) matrices, J. Algebra, 87, 2, 342-359 (1984) · Zbl 0537.16013
[23] Sims, Charles C., Computation with Finitely Presented Groups, Encyclopedia Math. Appl., vol. 48 (1994), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0828.20001
[24] Suzuki, Michio, Group Theory. I, Grundlehren Math. Wiss., vol. 247 (1982), Springer-Verlag: Springer-Verlag Berlin, translated from the Japanese by the author · Zbl 0472.20001
[25] Vogt, H., Sur les invariants fondamentaux des équations différentielles linéaires du second ordre, Ann. Sci. Ec. Norm. Super., 3, 6, 3-71 (1889) · JFM 21.0314.01
[26] Whittemore, Alice, On special linear characters of free groups of rank \(n \geq 4\), Proc. Amer. Math. Soc., 40, 383-388 (1973) · Zbl 0248.20031
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