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1,425
result(s) for
"Hyperbolic Geometry"
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Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces
In this paper, we provide a complete theory of Diophantine approximation in the limit set of a group acting on a Gromov hyperbolic
metric space. This summarizes and completes a long line of results by many authors, from Patterson’s classic ’76 paper to more recent
results of Hersonsky and Paulin (’02, ’04, ’07). Concrete examples of situations we consider which have not been considered before
include geometrically infinite Kleinian groups, geometrically finite Kleinian groups where the approximating point is not a fixed point
of any element of the group, and groups acting on infinite-dimensional hyperbolic space. Moreover, in addition to providing much greater
generality than any prior work of which we are aware, our results also give new insight into the nature of the connection between
Diophantine approximation and the geometry of the limit set within which it takes place. Two results are also contained here which are
purely geometric: a generalization of a theorem of Bishop and Jones (’97) to Gromov hyperbolic metric spaces, and a proof that the
uniformly radial limit set of a group acting on a proper geodesic Gromov hyperbolic metric space has zero Patterson–Sullivan measure
unless the group is quasiconvex-cocompact. The latter is an application of a Diophantine theorem.
eBook
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 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).
Journal Article
Complex hyperbolic triangle groups of type m,m,0;n1,n2,2
In this paper we study discreteness of complex hyperbolic triangle groups of type
[
m
,
m
,
0
;
n
1
,
n
2
,
2
]
, i.e. groups of isometries of the complex hyperbolic plane generated by three complex reflections of orders
n
1
,
n
2
,
2
in complex geodesics with pairwise distances
m
,
m
, 0. For fixed
m
, the parameter space of such groups is of real dimension one. We determine the possible orders for
n
1
and
n
2
and also intervals in the parameter space that correspond to discrete and non-discrete triangle groups.
Journal Article
Mirror stabilizers for lattice complex hyperbolic triangle groups
For each lattice complex hyperbolic triangle group, we study the Fuchsian stabilizers of (reprentatives of each group orbit of) mirrors of complex reflections. We give explicit generators for the stabilizers, and compute their signature in the sense of Fuchsian groups. For some groups, we also find explicit pairs of complex lines such that the union of their stabilizers generate the ambient lattice.
Journal Article
Analytic hyperbolic geometry and Albert Einstein's special theory of relativity
This book presents a powerful way to study Einstein's special theory of relativity and its underlying hyperbolic geometry in which analogies with classical results form the right tool. It introduces the notion of vectors into analytic hyperbolic geometry, where they are called gyrovectors.
eBook
Analytic hyperbolic geometry
This is the first book on analytic hyperbolic geometry, fully analogous to analytic Euclidean geometry. Analytic hyperbolic geometry regulates relativistic mechanics just as analytic Euclidean geometry regulates classical mechanics. The book presents a novel gyrovector space approach to analytic hyperbolic geometry, fully analogous to the well-known vector space approach to Euclidean geometry. A gyrovector is a hyperbolic vector. Gyrovectors are equivalence classes of directed gyrosegments that add according to the gyroparallelogram law just as vectors are equivalence classes of directed segments that add according to the parallelogram law. In the resulting “gyrolanguage” of the book one attaches the prefix “gyro” to a classical term to mean the analogous term in hyperbolic geometry. The prefix stems from Thomas gyration, which is the mathematical abstraction of the relativistic effect known as Thomas precession. Gyrolanguage turns out to be the language one needs to articulate novel analogies that the classical and the modern in this book share.
eBook
Sources of Hyperbolic Geometry
This book presents, for the first time in English, the papers of Beltrami, Klein, and Poincaré that brought hyperbolic geometry into the mainstream of mathematics. A recognition of Beltrami comparable to that given the pioneering works of Bolyai and Lobachevsky seems long overdue-not only because Beltrami rescued hyperbolic geometry from oblivion by proving it to be logically consistent, but because he gave it a concrete meaning (a model) that made hyperbolic geometry part of ordinary mathematics. The models subsequently discovered by Klein and Poincaré brought hyperbolic geometry even further down to earth and paved the way for the current explosion of activity in low-dimensional geometry and topology. By placing the works of these three mathematicians side by side and providing commentaries, this book gives the student, historian, or professional geometer a bird's-eye view of one of the great episodes in mathematics. The unified setting and historical context reveal the insights of Beltrami, Klein, and Poincaré in their full brilliance.
eBook
Coregularity of Fano varieties
The absolute regularity of a Fano variety, denoted by
reg
^
(
X
)
, is the largest dimension of the dual complex of a log Calabi–Yau structure on
X
. The absolute coregularity is defined to be
coreg
^
(
X
)
:
=
dim
X
-
reg
^
(
X
)
-
1
.
The coregularity is the complementary dimension of the regularity. We expect that the coregularity of a Fano variety governs, to a large extent, the geometry of
X
. In this note, we review the history of Fano varieties, give some examples, survey some theorems, introduce the coregularity, and propose several problems regarding this invariant of Fano varieties.
Journal Article
Finite-volume hyperbolic 3-manifolds are almost determined by their finite quotient groups
For any orientable finite-volume hyperbolic 3-manifold, this paper proves that the profinite isomorphism type of the fundamental group uniquely determines the isomorphism type of the first integral cohomology, as marked with the Thurston norm and the fibered classes; moreover, up to finite ambiguity, the profinite isomorphism type determines the isomorphism type of the fundamental group, among the class of finitely generated 3-manifold groups.
Journal Article