Generic subgroups of Aut 𝔹 n

Chiara de Fabritiis

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2002)

  • Volume: 1, Issue: 4, page 851-868
  • ISSN: 0391-173X

Abstract

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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 ) .

How to cite

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de Fabritiis, Chiara. "Generic subgroups of Aut $\mathbb {B}^n$." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 1.4 (2002): 851-868. <http://eudml.org/doc/84489>.

@article{deFabritiis2002,
abstract = {We prove that for a parabolic subgroup $\Gamma $ of $\{\rm Aut\} \mathbb \{B\}^n $ the fixed points sets of all elements in $\Gamma \setminus \lbrace \{\rm id\}_\{\{\mathbb \{B\}\}^n\}\rbrace $ are the same. This result, together with a deep study of the structure of subgroups of $\{\rm Aut\} \mathbb \{B\}^n $ acting freely and properly discontinuously on $ \mathbb \{B\}^n $, entails a generalization of the so called weak Hurwitz’s theorem: namely that, given a complex manifold $X$ covered by $ \mathbb \{B\}^n $ and such that the group of deck transformations of the covering is “sufficiently generic”, then $\{\rm id\}_X$ is isolated in $\{\rm Hol\}(X,X)$.},
author = {de Fabritiis, Chiara},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {4},
pages = {851-868},
publisher = {Scuola normale superiore},
title = {Generic subgroups of Aut $\mathbb \{B\}^n$},
url = {http://eudml.org/doc/84489},
volume = {1},
year = {2002},
}

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
PB - Scuola normale superiore
VL - 1
IS - 4
SP - 851
EP - 868
AB - We prove that for a parabolic subgroup $\Gamma $ of ${\rm Aut} \mathbb {B}^n $ the fixed points sets of all elements in $\Gamma \setminus \lbrace {\rm id}_{{\mathbb {B}}^n}\rbrace $ are the same. This result, together with a deep study of the structure of subgroups of ${\rm Aut} \mathbb {B}^n $ acting freely and properly discontinuously on $ \mathbb {B}^n $, entails a generalization of the so called weak Hurwitz’s theorem: namely that, given a complex manifold $X$ covered by $ \mathbb {B}^n $ and such that the group of deck transformations of the covering is “sufficiently generic”, then ${\rm id}_X$ is isolated in ${\rm Hol}(X,X)$.
LA - eng
UR - http://eudml.org/doc/84489
ER -

References

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  1. [1] M. Abate:, “Iteration theory of holomorphic maps on taut manifolds”, Mediterranean Press, Rende, Cosenza 1989. Zbl0747.32002MR1098711
  2. [2] M. Abate – J.-P. Vigué, Common fixed points in hyperbolic Riemann surfaces and convex domains, Proc. Amer. Math. Soc. 112 (1991), 503-512. Zbl0724.32012MR1065938
  3. [3] F. Bracci, Common fixed points of commuting holomorphic maps in the unit ball of n , Proc. Amer. Math. Soc. 127, 4, (1999), 1133-1141. Zbl0916.58006MR1610920
  4. [4] C. de Fabritiis, Commuting holomorphic functions and hyperbolic automorphisms, Proc. Amer. Math. Soc. 124 (1996), 3027-3037. Zbl0866.32004MR1371120
  5. [5] C. de Fabritiis – G. Gentili, On holomorphic maps which commute with hyperbolic automorphisms, Adv. Math. 144 (1999), 119-136. Zbl0976.32014MR1695235
  6. [6] C. de Fabritiis – A. Iannuzzi, Quotients of the unit ball of n by a free action of , Jour. d’Anal. Math. 85 (2001), 213-224. Zbl1008.32011MR1869609
  7. [7] H. M. Farkas – I. Kra, “Riemann Surfaces”, Springer, Berlin, 1980. Zbl0475.30001MR583745
  8. [8] T. Franzoni – E. Vesentini, “Holomorphic maps and invariant distances”, North-Holland, Amsterdam, 1980. Zbl0447.46040MR563329
  9. [9] M. H. Heins, A generalization of the Aumann-Carathéodory Starrheitssatz, Duke Math. J. 8 (1941), 312-316. Zbl67.0283.02MR4912JFM67.0283.02
  10. [10] A. Hurwitz, Über algebraische Gebilde mit eindeutingen Transformationen in sich, Math. Ann. 41 (1893), 403-442. MR1510753JFM24.0380.02
  11. [11] I. Kra, “Automorphic forms and Kleinian groups”, Beinjamin, New York, 1972. Zbl0253.30015MR357775
  12. [12] B. MacCluer, Iterates of holomorphic self-maps of the unit ball of n , Mich. Math. J. 30 (1983), 97-106. Zbl0528.32019MR694933
  13. [13] W. Rudin, “Function theory in the Unit Ball of n ”, Springer, Berlin 1980. Zbl0495.32001MR601594

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