zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Algebraic convergence theorems of complex Kleinian groups. (English) Zbl 1262.30041

Let 𝔾 be the n-dimensional sense-preserving Möbius group M( ¯ n ) or the unitary group U(1,n;). Let {G r,i } be a sequence of subgroups in the group 𝔾 generated by g 1,i ,g 2,i ,,g r,i , where r=1,2,.... If for each t (1tr), g t,i g t 𝔾 as i, then we say that {G r,i } algebraically converges to G r =g 1 ,g 2 ,,g r .

Let us suppose that for each i, the group G r,i is a Kleinian group. When is G r a Kleinian group? This problem has been studied by several authors: [T. Jørgensen and P. Klein, Quart. J. Math. 33, 325–332 (1982; Zbl 0499.30033); G. J. Martin, Acta Math. 163, No. 3–4, 253–289 (1989; Zbl 0698.20037); X. Wang, Isr. J. Math. 162, 221–233 (2007; Zbl 1161.20047)].

In the paper under review two algebraic convergence theorems are obtained. Let G be a subgroup of U(1,n;) containing a loxodromic element. Denote by W(G) the set

fh(G) G fix(f) ,

where h(G) is the set of all loxodromic elements in G and

G fix(f) =g G : fix ( f ) fix ( g )·

The author proves in the first theorem that given a sequence {G r,i } of subgroups of U(1,n;), if each G r,i is discrete, then the algebraic limit group G r is either a complex Kleinian group, or it is elementary, or W(G r ) is not finite.

In the second theorem it is proved that if the groups G r,i are complex Kleinian groups and {G r,i } satisfies the IP-condition, then the algebraic limit group G r is a complex Kleinian group, where we say that {G r,i } satisfies the IP-condition if the following conditions are satisfied: for any sequence {f i k } (f i k G r,i k ), if cardfix ( f i k )= for each k, and f i k f as k with f being the identity or parabolic, then {f i k } has uniformly bounded torsion.

MSC:
30F40Kleinian groups
20H10Fuchsian groups and their generalizations (group theory)