zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Ford and Dirichlet domains for cyclic subgroups of $\text{PSL}_2({\Bbb C})$ acting on ${{\Bbb H}^3_{\Bbb R}}$ and ${\partial}{{\Bbb H}^3_{\Bbb R}}$. (English) Zbl 0999.30028
Summary: Let $\Gamma$ be a cyclic subgroup of $\text{PSL}_2({\Bbb C})$ generated by a loxodromic element. The Ford and Dirichlet fundamental domains for the action of $\Gamma$ on ${{\Bbb H}^3_{\Bbb R}}$ are the complements of configurations of half-balls centered on the plane at infinity ${\partial}{{\Bbb H}^3_{\Bbb R}}$. {\it T. Jørgensen} [Math. Scand. 33, 250-260 (1973; Zbl 0286.30017)] proved that the boundary of the intersection of the Ford fundamental domain with ${\partial}{{\Bbb H}^3_{\Bbb R}}$ always consists of either two, four, or six circular arcs and stated that an arbitrarily large number of hemispheres could contribute faces to the Ford domain in the interior of ${{\Bbb H}^3_{\Bbb R}}$. We give new proofs of Jørgensen’s results, prove analogous facts for Dirichlet domains and for Ford and Dirichlet domains in the interior of ${{\Bbb H}^3_{\Bbb R}}$, and give a complete decomposition of the parameter space by the combinatorial type of the corresponding fundamental domain.

30F40Kleinian groups
20H10Fuchsian groups and their generalizations (group theory)
Full Text: DOI