no. 4
Generic subgroups of Aut 𝔹 n
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 1 (2002) no. 4, pp. 851-868

We prove that for a parabolic subgroup Γ of Aut 𝔹 n the fixed points sets of all elements in Γ∖{ id 𝔹 n } are the same. This result, together with a deep study of the structure of subgroups of Aut 𝔹 n acting freely and properly discontinuously on 𝔹 n , entails a generalization of the so called weak Hurwitz’s theorem: namely that, given a complex manifold X covered by 𝔹 n and such that the group of deck transformations of the covering is “sufficiently generic”, then id X is isolated in Hol (X,X).

Classification : 32A10, 32A40, 32H15, 32A30 
@article{ASNSP_2002_5_1_4_851_0,
     author = {de Fabritiis, Chiara},
     title = {Generic subgroups of {Aut} $\mathbb {B}^n$},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {851--868},
     year = {2002},
     publisher = {Scuola normale superiore},
     volume = {Ser. 5, 1},
     number = {4},
     mrnumber = {1991005},
     language = {en},
     url = {https://www.numdam.org/item/ASNSP_2002_5_1_4_851_0/}
}
TY  - JOUR
AU  - de Fabritiis, Chiara
TI  - Generic subgroups of Aut $\mathbb {B}^n$
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2002
SP  - 851
EP  - 868
VL  - 1
IS  - 4
PB  - Scuola normale superiore
UR  - https://www.numdam.org/item/ASNSP_2002_5_1_4_851_0/
LA  - en
ID  - ASNSP_2002_5_1_4_851_0
ER  - 
%0 Journal Article
%A de Fabritiis, Chiara
%T Generic subgroups of Aut $\mathbb {B}^n$
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2002
%P 851-868
%V 1
%N 4
%I Scuola normale superiore
%U https://www.numdam.org/item/ASNSP_2002_5_1_4_851_0/
%G en
%F ASNSP_2002_5_1_4_851_0
de Fabritiis, Chiara. Generic subgroups of Aut $\mathbb {B}^n$. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 1 (2002) no. 4, pp. 851-868. https://www.numdam.org/item/ASNSP_2002_5_1_4_851_0/

[1] M. Abate:, “Iteration theory of holomorphic maps on taut manifolds”, Mediterranean Press, Rende, Cosenza 1989. | Zbl | MR

[2] M. Abate - J.-P. Vigué, Common fixed points in hyperbolic Riemann surfaces and convex domains, Proc. Amer. Math. Soc. 112 (1991), 503-512. | Zbl | MR

[3] F. Bracci, Common fixed points of commuting holomorphic maps in the unit ball of ℂ n , Proc. Amer. Math. Soc. 127, 4, (1999), 1133-1141. | Zbl | MR

[4] C. De Fabritiis, Commuting holomorphic functions and hyperbolic automorphisms, Proc. Amer. Math. Soc. 124 (1996), 3027-3037. | Zbl | MR

[5] C. De Fabritiis - G. Gentili, On holomorphic maps which commute with hyperbolic automorphisms, Adv. Math. 144 (1999), 119-136. | Zbl | MR

[6] C. De Fabritiis - A. Iannuzzi, Quotients of the unit ball of ℂ n by a free action of ℤ, Jour. d'Anal. Math. 85 (2001), 213-224. | Zbl | MR

[7] H. M. Farkas - I. Kra, “Riemann Surfaces”, Springer, Berlin, 1980. | Zbl | MR

[8] T. Franzoni - E. Vesentini, “Holomorphic maps and invariant distances”, North-Holland, Amsterdam, 1980. | Zbl | MR

[9] M. H. Heins, A generalization of the Aumann-Carathéodory Starrheitssatz, Duke Math. J. 8 (1941), 312-316. | MR | JFM

[10] A. Hurwitz, Über algebraische Gebilde mit eindeutingen Transformationen in sich, Math. Ann. 41 (1893), 403-442. | MR | JFM

[11] I. Kra, “Automorphic forms and Kleinian groups”, Beinjamin, New York, 1972. | Zbl | MR

[12] B. Maccluer, Iterates of holomorphic self-maps of the unit ball of ℂ n , Mich. Math. J. 30 (1983), 97-106. | Zbl | MR

[13] W. Rudin, “Function theory in the Unit Ball of ℂ n ”, Springer, Berlin 1980. | Zbl | MR