On the number of factors in the unipotent factorization of holomorphic mappings into 𝑆𝐿₂(ℂ)

Ivarsson, Björn; Kutzschebauch, Frank

doi:10.1090/S0002-9939-2011-11025-6

On the number of factors in the unipotent factorization of holomorphic mappings into $\text {SL}_2(\mathbb {C})$
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by Björn Ivarsson and Frank Kutzschebauch

Proc. Amer. Math. Soc. 140 (2012), 823-838

DOI: https://doi.org/10.1090/S0002-9939-2011-11025-6

Published electronically: June 23, 2011

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Abstract:

We estimate the number of unipotent elements needed to factor a null-homotopic holomorphic map from a finite dimensional reduced Stein space $X$ into $\text {SL}_2(\mathbb {C})$.

References

P. M. Cohn, On the structure of the $\textrm {GL}_{2}$ of a ring, Inst. Hautes Études Sci. Publ. Math. 30 (1966), 5–53. MR 207856
Franc Forstnerič, The Oka principle for sections of stratified fiber bundles, Pure Appl. Math. Q. 6 (2010), no. 3, Special Issue: In honor of Joseph J. Kohn., 843–874. MR 2677316, DOI 10.4310/PAMQ.2010.v6.n3.a11
Franc Forstnerič and Jasna Prezelj, Extending holomorphic sections from complex subvarieties, Math. Z. 236 (2001), no. 1, 43–68. MR 1812449, DOI 10.1007/PL00004826
Franc Forstnerič and Jasna Prezelj, Oka’s principle for holomorphic submersions with sprays, Math. Ann. 322 (2002), no. 4, 633–666. MR 1905108, DOI 10.1007/s002080100249
M. Gromov, Oka’s principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2 (1989), no. 4, 851–897. MR 1001851, DOI 10.1090/S0894-0347-1989-1001851-9
Björn Ivarsson and Frank Kutzschebauch. Holomorphic factorization of mappings into $\mbox {SL}_n(\mathbb {C})$. Submitted for publication, arXiv:0812.0312, December 2008.
Björn Ivarsson and Frank Kutzschebauch, A solution of Gromov’s Vaserstein problem, C. R. Math. Acad. Sci. Paris 346 (2008), no. 23-24, 1239–1243 (English, with English and French summaries). MR 2473300, DOI 10.1016/j.crma.2008.10.017
Björn Ivarsson and Frank Kutzschebauch. On Kazhdan’s property (T) for the special linear group of holomorphic functions, 2010, unpublished manuscript.
A. A. Suslin, The structure of the special linear group over rings of polynomials, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 2, 235–252, 477 (Russian). MR 0472792
L. N. Vaserstein, Reduction of a matrix depending on parameters to a diagonal form by addition operations, Proc. Amer. Math. Soc. 103 (1988), no. 3, 741–746. MR 947649, DOI 10.1090/S0002-9939-1988-0947649-X
Wilberd van der Kallen, $\textrm {SL}_{3}(\textbf {C}[X])$ does not have bounded word length, Algebraic $K$-theory, Part I (Oberwolfach, 1980) Lecture Notes in Math., vol. 966, Springer, Berlin-New York, 1982, pp. 357–361. MR 689383