James W. Cannon
Encyclopedia
James W. Cannon is an American mathematician working in the areas of low-dimensional topology
and geometric group theory
. He is an Orson Pratt Professor of Mathematics at Brigham Young University
.
, Pennsylvania
. Cannon received a PhD in Mathematics from the University of Utah
in 1969, under the direction of Cecil Burgess.
He was a Professor at the University of Wisconsin, Madison from 1977 to 1985. In 1986 Cannon was appointed an Orson Pratt Professor of Mathematics at Brigham Young University
.
Cannon gave an AMS Invited address at the meeting of the American Mathematical Society
in Seattle in August 1977, an ICM Invited Address at the International Congress of Mathematicians
in Helsinki 1978, and delivered the 1982 Mathematical Association of America
Hedrick Lectures in Toronto, Canada.
Cannon was elected to the American Mathematical Society
Council in 2003 with the term of service February 1, 2004, to January 31, 2007.
In 1993 Cannon delivered the 30-th annual Karl G. Maeser Distinguished Faculty Lecture at Brigham Young University
.
James Cannon is a devout member of the LDS Church.
His first famous result came in late 70s when Cannon gave a complete solution to a long-standing "double suspension" problem posed by John Milnor
. Cannon proved that the double suspension of a homology sphere
is a topological sphere. R. D. Edwards had previously proven this in many cases.
The results of Cannon's paper were used by Cannon, Bryant and Lacher to prove (1979) an important case of the so-called characterization conjecture for topological manifolds. The conjecture says that a generalized n-manifold M, where n ≥ 5, which satisfies the "disjoint disk property" is a topological manifold. Cannon, Bryant and Lacher established that the conjecture holds under the assumption that M be a manifold except possibly at a set of dimension (n−2)/2. Later Quinn completed the proof that the characterization conjecture holds in full generality.
s, hyperbolic geometry
and Kleinian group
s and he is considered one of the key figures in the birth of geometric group theory
as a distinct subject in late 1980s and early 1990s. Cannon's 1984 paper "The combinatorial structure of cocompact discrete hyperbolic groups" was one of the forerunners in the development of the theory of word-hyperbolic groups, a notion that was introduced and developed three years later in a seminal 1987 monograph of Gromov. Cannon's paper explored combinatorial and algorithmic aspects of the Cayley graph
s of Kleinian groups and related them to the geometric features of the actions of these groups on the hyperbolic space. In particular, Cannon proved that convex-cocompact Kleinian groups admit finite presentations where the Dehn algorithm solves the word problem
. The latter condition later turned out to give one of equivalent characterization of being word-hyperbolic and, moreover, Cannon's original proof essentially went through without change to show that the word problem in word-hyperbolic groups is solvable by Dehn's algorithm. Cannon's 1984 paper also introduced an important notion a cone type of an element of a finitely generated group (roughly, the set of all geodesic extensions of an element). Cannon proved that a convex-cocompact Kleinian group has only finitely many cone types (with respect to a fixed finite generating set of that group) and showed how to use this fact to conclude that the growth series of the group is a rational function
. These arguments also turned out to generalize to the word-hyperbolic group context. Now standard proofs of the fact that the set of geodesic words in a word-hyperbolic group is a regular language
also use finiteness of the number of cone types.
Cannon's work also introduced an important notion of almost convexity for Cayley graphs of finitely generated groups, a notion that led to substantial further study and generalizations.
An influential paper of Cannon and Thurston
"Group invariant Peano curves", that first circulated in a preprint form in mid-1980s, introduced the notion of what is now called the Cannon–Thurston map. They considered the case of a closed hyperbolic 3-manifold M that fibers
over the circle with the fiber being a closed hyperbolic surface S. In this case the universal cover of S, which is identified with the hyperbolic plane
, admits an embedding into the universal cover of M, which is the hyperbolic 3-space. Cannon and Thurston proved that this embedding extends to a continuous π1(S)-equivariant surjective map (now called the Cannon–Thurston map) from the ideal boundary of the hyperbolic plane (the circle) to the ideal boundary of the hyperbolic 3-space (the 2-sphere).
Although the paper of Cannon and Thurston was finally published only in 2007, in the meantime it has generated considerable further research and a number of significant generalizations (both in the contexts of Kleinian groups and of word-hyperbolic groups), including the work of Mahan Mitra
, Klarreich, Bowditch and others.
s. The theory of automatic groups brought new computational ideas from computer science
to geometric group theory
and played an important role in the development of the subject in 1990s.
A 1994 paper of Cannon gave a proof of the "combinatorial Riemann mapping theorem" that was motivated by the classic Riemann mapping theorem in complex analysis
. The goal was to understand when an action
of a group by homeomorphism
s on a 2-sphere is (up to a topological conjugation) an action on the standard Riemann sphere
by Möbius transformations. The "combinatorial Riemann mapping theorem" of Cannon gave a set of sufficient conditions when a sequence of finer and finer combinatorial subdivisions of a topological surface determine, in the appropriate sense and after passing to the limit, an actual conformal structure on that surface. This paper of Cannon led to an important conjecture, first explicitly formulated by Cannon and Swenson in 1998 (but also suggested in implicit form in Section 8 of Cannon's 1994 paper) and now known as Cannon's conjecture, regarding characterizing word-hyperbolic groups with the 2-sphere as the boundary. The conjecture (Conjecture 5.1 in ) states that if the ideal boundary of a word-hyperbolic group G is homeomorphic
to the 2-sphere, then G admits a properly discontinuous cocompact isometric action on the hyperbolic 3-space (so that G is essentially a 3-dimensional Kleinian group
). In analytic terms Cannon's conjecture is equivalent to saying that if the ideal boundary of a word-hyperbolic group G is homeomorphic to the 2-sphere then this boundary, with the visual metric coming from the Cayley graph
of G, is quasisymmetric
to the standard 2-sphere.
The 1998 paper of Cannon and Swenson gave an initial approach to this conjecture by proving that the conjecture holds under an extra assumption that the family of standard "disks" in the boundary of the group satisfies a combinatorial "conformal" property. The main result of Cannon's 1994 paper played a key role in the proof. This approach to Cannon's conjecture and related problems was pushed further later in the joint work of Cannon, Floyd and Parry.
Cannon's conjecture motivated much of subsequent work by other mathematicians and to a substantial degree informed subsequent interaction between geometric group theory
and the theory of analysis on metric spaces. Cannon's conjecture was motivated (see ) by Thurston's Geometrization Conjecture
and by trying to understand why in dimension three variable negative curvature can be promoted to constant negative curvature. Although the Geometrization conjecture
was recently settled by Perelman
, Cannon's conjecture remains wide open and is considered one of the key outstanding open problems in geometric group theory
and geometric topology
.
s. In particular they suggested (see section 3.4 of ) that such "negatively curved" local structure is manifested in highly folded and highly connected nature of the brain and the lung tissue.
Low-dimensional topology
In mathematics, low-dimensional topology is the branch of topology that studies manifolds of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. It can be regarded as a part of geometric topology.A number of...
and geometric group theory
Geometric group theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act .Another important...
. He is an Orson Pratt Professor of Mathematics at Brigham Young University
Brigham Young University
Brigham Young University is a private university located in Provo, Utah. It is owned and operated by The Church of Jesus Christ of Latter-day Saints , and is the United States' largest religious university and third-largest private university.Approximately 98% of the university's 34,000 students...
.
Biographical data
James W. Cannon was born on January 30, 1943, in BellefonteBellefonte, Pennsylvania
Bellefonte is a borough in Centre County, Pennsylvania, United States. It lies about twelve miles northeast of State College and is part of the State College, Pennsylvania Metropolitan Statistical Area...
, Pennsylvania
Pennsylvania
The Commonwealth of Pennsylvania is a U.S. state that is located in the Northeastern and Mid-Atlantic regions of the United States. The state borders Delaware and Maryland to the south, West Virginia to the southwest, Ohio to the west, New York and Ontario, Canada, to the north, and New Jersey to...
. Cannon received a PhD in Mathematics from the University of Utah
University of Utah
The University of Utah, also known as the U or the U of U, is a public, coeducational research university in Salt Lake City, Utah, United States. The university was established in 1850 as the University of Deseret by the General Assembly of the provisional State of Deseret, making it Utah's oldest...
in 1969, under the direction of Cecil Burgess.
He was a Professor at the University of Wisconsin, Madison from 1977 to 1985. In 1986 Cannon was appointed an Orson Pratt Professor of Mathematics at Brigham Young University
Brigham Young University
Brigham Young University is a private university located in Provo, Utah. It is owned and operated by The Church of Jesus Christ of Latter-day Saints , and is the United States' largest religious university and third-largest private university.Approximately 98% of the university's 34,000 students...
.
Cannon gave an AMS Invited address at the meeting of the American Mathematical Society
American Mathematical Society
The American Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians.The society is one of the...
in Seattle in August 1977, an ICM Invited Address at the International Congress of Mathematicians
International Congress of Mathematicians
The International Congress of Mathematicians is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union ....
in Helsinki 1978, and delivered the 1982 Mathematical Association of America
Mathematical Association of America
The Mathematical Association of America is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure and applied mathematicians; computer scientists;...
Hedrick Lectures in Toronto, Canada.
Cannon was elected to the American Mathematical Society
American Mathematical Society
The American Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians.The society is one of the...
Council in 2003 with the term of service February 1, 2004, to January 31, 2007.
In 1993 Cannon delivered the 30-th annual Karl G. Maeser Distinguished Faculty Lecture at Brigham Young University
Brigham Young University
Brigham Young University is a private university located in Provo, Utah. It is owned and operated by The Church of Jesus Christ of Latter-day Saints , and is the United States' largest religious university and third-largest private university.Approximately 98% of the university's 34,000 students...
.
James Cannon is a devout member of the LDS Church.
Early work
Cannon's early work concerned topological aspects of embedded surfaces in R3 and understanding the difference between "tame" and "wild" surfaces.His first famous result came in late 70s when Cannon gave a complete solution to a long-standing "double suspension" problem posed by John Milnor
John Milnor
John Willard Milnor is an American mathematician known for his work in differential topology, K-theory and dynamical systems. He won the Fields Medal in 1962, the Wolf Prize in 1989, and the Abel Prize in 2011. Milnor is a distinguished professor at Stony Brook University...
. Cannon proved that the double suspension of a homology sphere
Homology sphere
In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer n ≥ 1. That is,andTherefore X is a connected space, with one non-zero higher Betti number: bn...
is a topological sphere. R. D. Edwards had previously proven this in many cases.
The results of Cannon's paper were used by Cannon, Bryant and Lacher to prove (1979) an important case of the so-called characterization conjecture for topological manifolds. The conjecture says that a generalized n-manifold M, where n ≥ 5, which satisfies the "disjoint disk property" is a topological manifold. Cannon, Bryant and Lacher established that the conjecture holds under the assumption that M be a manifold except possibly at a set of dimension (n−2)/2. Later Quinn completed the proof that the characterization conjecture holds in full generality.
1980s: Hyperbolic geometry, 3-manifolds and geometric group theory
In 1980s the focus of Cannon's work shifted to the study of 3-manifold3-manifold
In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...
s, hyperbolic geometry
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
and Kleinian group
Kleinian group
In mathematics, a Kleinian group is a discrete subgroup of PSL. The group PSL of 2 by 2 complex matrices of determinant 1 modulo its center has several natural representations: as conformal transformations of the Riemann sphere, and as orientation-preserving isometries of 3-dimensional hyperbolic...
s and he is considered one of the key figures in the birth of geometric group theory
Geometric group theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act .Another important...
as a distinct subject in late 1980s and early 1990s. Cannon's 1984 paper "The combinatorial structure of cocompact discrete hyperbolic groups" was one of the forerunners in the development of the theory of word-hyperbolic groups, a notion that was introduced and developed three years later in a seminal 1987 monograph of Gromov. Cannon's paper explored combinatorial and algorithmic aspects of the Cayley graph
Cayley graph
In mathematics, a Cayley graph, also known as a Cayley colour graph, Cayley diagram, group diagram, or colour group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem and uses a specified, usually finite, set of generators for the group...
s of Kleinian groups and related them to the geometric features of the actions of these groups on the hyperbolic space. In particular, Cannon proved that convex-cocompact Kleinian groups admit finite presentations where the Dehn algorithm solves the word problem
Word problem for groups
In mathematics, especially in the area of abstract algebra known as combinatorial group theory, the word problem for a finitely generated group G is the algorithmic problem of deciding whether two words in the generators represent the same element...
. The latter condition later turned out to give one of equivalent characterization of being word-hyperbolic and, moreover, Cannon's original proof essentially went through without change to show that the word problem in word-hyperbolic groups is solvable by Dehn's algorithm. Cannon's 1984 paper also introduced an important notion a cone type of an element of a finitely generated group (roughly, the set of all geodesic extensions of an element). Cannon proved that a convex-cocompact Kleinian group has only finitely many cone types (with respect to a fixed finite generating set of that group) and showed how to use this fact to conclude that the growth series of the group is a rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
. These arguments also turned out to generalize to the word-hyperbolic group context. Now standard proofs of the fact that the set of geodesic words in a word-hyperbolic group is a regular language
Regular language
In theoretical computer science and formal language theory, a regular language is a formal language that can be expressed using regular expression....
also use finiteness of the number of cone types.
Cannon's work also introduced an important notion of almost convexity for Cayley graphs of finitely generated groups, a notion that led to substantial further study and generalizations.
An influential paper of Cannon and Thurston
William Thurston
William Paul Thurston is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds...
"Group invariant Peano curves", that first circulated in a preprint form in mid-1980s, introduced the notion of what is now called the Cannon–Thurston map. They considered the case of a closed hyperbolic 3-manifold M that fibers
Fiber bundle
In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...
over the circle with the fiber being a closed hyperbolic surface S. In this case the universal cover of S, which is identified with the hyperbolic plane
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
, admits an embedding into the universal cover of M, which is the hyperbolic 3-space. Cannon and Thurston proved that this embedding extends to a continuous π1(S)-equivariant surjective map (now called the Cannon–Thurston map) from the ideal boundary of the hyperbolic plane (the circle) to the ideal boundary of the hyperbolic 3-space (the 2-sphere).
Although the paper of Cannon and Thurston was finally published only in 2007, in the meantime it has generated considerable further research and a number of significant generalizations (both in the contexts of Kleinian groups and of word-hyperbolic groups), including the work of Mahan Mitra
Mahan Mitra
Mahan Mitra, also known as Mahan Maharaj and Swami Vidyanathananda, is an Indian mathematician and a recipient of the 2011 Shanti Swarup Bhatnagar Award in Mathematical Sciences..- Early education :...
, Klarreich, Bowditch and others.
1990s and 2000s: Automatic groups, discrete conformal geometry and Cannon's conjecture
Cannon was one of the co-authors of the 1992 book "Word Processing in Groups" which introduced, formalized and developed the theory of automatic groupAutomatic group
In mathematics, an automatic group is a finitely generated group equipped with several finite-state automata. These automata can tell if a given word representation of a group element is in a "canonical form" and can tell if two elements given in canonical words differ by a generator.More...
s. The theory of automatic groups brought new computational ideas from computer science
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
to geometric group theory
Geometric group theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act .Another important...
and played an important role in the development of the subject in 1990s.
A 1994 paper of Cannon gave a proof of the "combinatorial Riemann mapping theorem" that was motivated by the classic Riemann mapping theorem in complex analysis
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
. The goal was to understand when an action
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
of a group by homeomorphism
Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
s on a 2-sphere is (up to a topological conjugation) an action on the standard Riemann sphere
Riemann sphere
In mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...
by Möbius transformations. The "combinatorial Riemann mapping theorem" of Cannon gave a set of sufficient conditions when a sequence of finer and finer combinatorial subdivisions of a topological surface determine, in the appropriate sense and after passing to the limit, an actual conformal structure on that surface. This paper of Cannon led to an important conjecture, first explicitly formulated by Cannon and Swenson in 1998 (but also suggested in implicit form in Section 8 of Cannon's 1994 paper) and now known as Cannon's conjecture, regarding characterizing word-hyperbolic groups with the 2-sphere as the boundary. The conjecture (Conjecture 5.1 in ) states that if the ideal boundary of a word-hyperbolic group G is homeomorphic
Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
to the 2-sphere, then G admits a properly discontinuous cocompact isometric action on the hyperbolic 3-space (so that G is essentially a 3-dimensional Kleinian group
Kleinian group
In mathematics, a Kleinian group is a discrete subgroup of PSL. The group PSL of 2 by 2 complex matrices of determinant 1 modulo its center has several natural representations: as conformal transformations of the Riemann sphere, and as orientation-preserving isometries of 3-dimensional hyperbolic...
). In analytic terms Cannon's conjecture is equivalent to saying that if the ideal boundary of a word-hyperbolic group G is homeomorphic to the 2-sphere then this boundary, with the visual metric coming from the Cayley graph
Cayley graph
In mathematics, a Cayley graph, also known as a Cayley colour graph, Cayley diagram, group diagram, or colour group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem and uses a specified, usually finite, set of generators for the group...
of G, is quasisymmetric
Quasisymmetric map
In mathematics, a quasisymmetric homeomorphism between metric spaces is a map that generalizes bi-Lipschitz maps. While bi-Lipschitz maps shrink or expand the diameter of a set by no more than a multiplicative factor, quasisymmetric maps satisfy the weaker geometric property that they preserve the...
to the standard 2-sphere.
The 1998 paper of Cannon and Swenson gave an initial approach to this conjecture by proving that the conjecture holds under an extra assumption that the family of standard "disks" in the boundary of the group satisfies a combinatorial "conformal" property. The main result of Cannon's 1994 paper played a key role in the proof. This approach to Cannon's conjecture and related problems was pushed further later in the joint work of Cannon, Floyd and Parry.
Cannon's conjecture motivated much of subsequent work by other mathematicians and to a substantial degree informed subsequent interaction between geometric group theory
Geometric group theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act .Another important...
and the theory of analysis on metric spaces. Cannon's conjecture was motivated (see ) by Thurston's Geometrization Conjecture
Geometrization conjecture
Thurston's geometrization conjecture states that compact 3-manifolds can be decomposed canonically into submanifolds that have geometric structures. The geometrization conjecture is an analogue for 3-manifolds of the uniformization theorem for surfaces...
and by trying to understand why in dimension three variable negative curvature can be promoted to constant negative curvature. Although the Geometrization conjecture
Geometrization conjecture
Thurston's geometrization conjecture states that compact 3-manifolds can be decomposed canonically into submanifolds that have geometric structures. The geometrization conjecture is an analogue for 3-manifolds of the uniformization theorem for surfaces...
was recently settled by Perelman
Grigori Perelman
Grigori Yakovlevich Perelman is a Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology.In 1992, Perelman proved the soul conjecture. In 2002, he proved Thurston's geometrization conjecture...
, Cannon's conjecture remains wide open and is considered one of the key outstanding open problems in geometric group theory
Geometric group theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act .Another important...
and geometric topology
Geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...
.
Applications to biology
The ideas of combinatorial conformal geometry that underlie Cannon's proof of the "combinatorial Riemann mapping theorem", were applied by Cannon, Floyd and Parry (2000) to the study of large-scale growth patterns of biological organisms. Cannon, Floyd and Parry produced a mathematical growth model which demonstrated that some systems determined by simple finite subdivision rules can results in objects (in their example, a tree trunk) whose large-scale form oscillates wildly over time even though the local subdivision laws remain the same. Cannon, Floyd and Parry also applied their model to the analysis of the growth patterns of rat tissue. They suggested that the "negatively curved" (or non-euclidean) nature of microscopic growth patterns of biological organisms is one of the key reasons why large-scale organisms do not look like crystals or polyhedral shapes but in fact in many cases resemble self-similar fractalFractal
A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
s. In particular they suggested (see section 3.4 of ) that such "negatively curved" local structure is manifested in highly folded and highly connected nature of the brain and the lung tissue.
See also
- Geometric group theoryGeometric group theoryGeometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act .Another important...
- Low-dimensional topologyLow-dimensional topologyIn mathematics, low-dimensional topology is the branch of topology that studies manifolds of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. It can be regarded as a part of geometric topology.A number of...
- Word-hyperbolic group
- Geometrization conjectureGeometrization conjectureThurston's geometrization conjecture states that compact 3-manifolds can be decomposed canonically into submanifolds that have geometric structures. The geometrization conjecture is an analogue for 3-manifolds of the uniformization theorem for surfaces...
- Hyperbolic manifoldHyperbolic manifoldIn mathematics, a hyperbolic n-manifold is a complete Riemannian n-manifold of constant sectional curvature -1.Every complete, connected, simply-connected manifold of constant negative curvature −1 is isometric to the real hyperbolic space Hn. As a result, the universal cover of any closed manifold...
- Kleinian groupKleinian groupIn mathematics, a Kleinian group is a discrete subgroup of PSL. The group PSL of 2 by 2 complex matrices of determinant 1 modulo its center has several natural representations: as conformal transformations of the Riemann sphere, and as orientation-preserving isometries of 3-dimensional hyperbolic...