The Menger curve and spherical CR uniformization of a closed hyperbolic 3-orbifold
Let G 6 , 3 be a hyperbolic polygon-group with boundary the Menger curve. Granier (Groupes discrets en géométrie hyperbolique—aspects effectifs, Université de Fribourg, 2015) constructed a discrete, convex cocompact and faithful representation ρ of G 6 , 3 into PU ( 2 , 1 ) . We show the 3-orbifold...
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Published in | Geometriae dedicata Vol. 218; no. 4 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
Dordrecht
Springer Netherlands
01.08.2024
Springer Nature B.V |
Subjects | |
Online Access | Get full text |
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Summary: | Let
G
6
,
3
be a hyperbolic polygon-group with boundary the Menger curve. Granier (Groupes discrets en géométrie hyperbolique—aspects effectifs, Université de Fribourg, 2015) constructed a discrete, convex cocompact and faithful representation
ρ
of
G
6
,
3
into
PU
(
2
,
1
)
. We show the 3-orbifold at infinity of
ρ
(
G
6
,
3
)
is a closed hyperbolic 3-orbifold, with underlying space the 3-sphere and singular locus the
Z
3
-coned chain-link
C
(
6
,
-
2
)
. This answers the second part of Kapovich’s Conjecture 10.6 in Kapovich (in: In the tradition of thurston II. Geometry and groups, Springer, Cham, 2022), and it also provides the second explicit example of a closed hyperbolic 3-orbifold that admits a uniformizable spherical CR-structure after Schwartz’s first example in Schwartz (Invent Math 151(2):221–295, 2003). |
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ISSN: | 0046-5755 1572-9168 |
DOI: | 10.1007/s10711-024-00934-y |