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''hyperbolic space in the sense of Gromov''
 
''hyperbolic space in the sense of Gromov''
  
This notion provides a uniform  "global"  approach to such objects as the hyperbolic plane, simply-connected Riemannian manifolds with pinched negative sectional curvature, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g1102402.png" />-spaces, and metric trees. Various  "hyperbolic properties"  introduced earlier (mostly in the context of group theory) [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]] were summed up and further developed by M. Gromov in his seminal paper [[#References|[a4]]]. More detailed expositions of (some parts of) Gromov's work can be found in [[#References|[a5]]] and [[#References|[a6]]].
+
This notion provides a uniform  "global"  approach to such objects as the hyperbolic plane, simply-connected Riemannian manifolds with pinched negative sectional curvature, $  { \mathop{\rm CAT} } ( - 1 ) $-
 +
spaces, and metric trees. Various  "hyperbolic properties"  introduced earlier (mostly in the context of group theory) [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]] were summed up and further developed by M. Gromov in his seminal paper [[#References|[a4]]]. More detailed expositions of (some parts of) Gromov's work can be found in [[#References|[a5]]] and [[#References|[a6]]].
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g1102403.png" /> is a [[Metric space|metric space]], then the Gromov product of two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g1102404.png" /> with respect to a reference point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g1102405.png" /> is defined as
+
If $  ( X,d ) $
 +
is a [[Metric space|metric space]], then the Gromov product of two points $  x,y \in X $
 +
with respect to a reference point $  o \in X $
 +
is defined as
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g1102406.png" /></td> </tr></table>
+
$$
 +
( x.y ) _ {o} = {
 +
\frac{1}{2}
 +
} [ d ( o,x ) + d ( o,y ) - d ( x,y ) ]
 +
$$
  
(when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g1102407.png" /> is a tree, this product coincides with the confluent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g1102408.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g1102409.png" />, i.e., the length of the common part of the geodesic segments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024011.png" />). The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024012.png" /> is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024014.png" />-hyperbolic (with a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024015.png" />) if the Gromov product satisfies the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024017.png" />-ultrametric inequality
+
(when $  X $
 +
is a tree, this product coincides with the confluent of $  x $
 +
and $  y $,  
 +
i.e., the length of the common part of the geodesic segments $  [ o,x ] $
 +
and $  [ o,y ] $).  
 +
The space $  X $
 +
is called $  \delta $-
 +
hyperbolic (with a constant $  \delta \geq  0 $)  
 +
if the Gromov product satisfies the $  \delta $-
 +
ultrametric inequality
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024018.png" /></td> </tr></table>
+
$$
 +
( x.y ) _ {o} \geq  \min  \{ ( x.z ) _ {o} , ( y.z ) _ {o} \} - \delta
 +
$$
  
for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024019.png" /> points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024020.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024021.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024022.png" />-hyperbolic for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024023.png" />, it is called Gromov hyperbolic. Any metric tree is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024024.png" />-hyperbolic, and, conversely, any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024025.png" />-hyperbolic space isometrically imbeds into a tree. For Cartan–Hadamard manifolds, hyperbolicity is equivalent to the strong visibility property [[#References|[a7]]].
+
for any $  4 $
 +
points $  o,x,y,z \in X $.  
 +
If $  X $
 +
is $  \delta $-
 +
hyperbolic for some $  \delta \geq  0 $,  
 +
it is called Gromov hyperbolic. Any metric tree is 0 $-
 +
hyperbolic, and, conversely, any 0 $-
 +
hyperbolic space isometrically imbeds into a tree. For Cartan–Hadamard manifolds, hyperbolicity is equivalent to the strong visibility property [[#References|[a7]]].
  
Often one adds to the definition of Gromov hyperbolicity the following two requirements: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024026.png" /> is geodesic (cf. also [[Geodesic manifold|Geodesic manifold]]), i.e., any two points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024027.png" /> can be joined by a path (a geodesic) whose length coincides with the distance between these points; and 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024028.png" /> is proper, i.e., all metric balls in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024029.png" /> are compact. Then hyperbolicity is equivalent to the following thin triangles condition: there exists a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024030.png" /> such that for any geodesic triangle in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024031.png" /> one can choose a point on each of the sides of the triangle in such a way that the pairwise distances between these <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024032.png" /> points are all not greater than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024033.png" />.
+
Often one adds to the definition of Gromov hyperbolicity the following two requirements: 1) $  X $
 +
is geodesic (cf. also [[Geodesic manifold|Geodesic manifold]]), i.e., any two points in $  X $
 +
can be joined by a path (a geodesic) whose length coincides with the distance between these points; and 2) $  X $
 +
is proper, i.e., all metric balls in $  X $
 +
are compact. Then hyperbolicity is equivalent to the following thin triangles condition: there exists a constant $  \delta  ^  \prime  $
 +
such that for any geodesic triangle in $  X $
 +
one can choose a point on each of the sides of the triangle in such a way that the pairwise distances between these $  3 $
 +
points are all not greater than $  \delta  ^  \prime  $.
  
Yet another definition of hyperbolicity can be given in terms of exponential divergence of geodesic rays in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024034.png" />, cf. [[#References|[a8]]].
+
Yet another definition of hyperbolicity can be given in terms of exponential divergence of geodesic rays in $  X $,  
 +
cf. [[#References|[a8]]].
  
A sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024035.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024036.png" /> is called convergent if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024037.png" />. The hyperbolic boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024038.png" /> of a hyperbolic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024039.png" /> is defined as the set of equivalence classes of convergent sequences with respect to the equivalence relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024040.png" />. Any geodesic ray in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024041.png" /> is convergent, and, conversely, for any class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024042.png" /> and any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024043.png" /> there exists a geodesic ray (not necessarily unique!) joining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024045.png" />, i.e., starting from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024046.png" /> and belonging to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024047.png" />. Equivalently, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024048.png" /> can be defined as the set of asymptotic classes of geodesic rays, with two rays being asymptotic if they are within bounded distance from each other.
+
A sequence $  ( x _ {n} ) $
 +
in $  X $
 +
is called convergent if $  ( x _ {n} .x _ {m} ) \rightarrow \infty $.  
 +
The hyperbolic boundary $  \partial  X $
 +
of a hyperbolic space $  X $
 +
is defined as the set of equivalence classes of convergent sequences with respect to the equivalence relation $  ( x _ {n} ) \sim ( y _ {n} ) \iff ( x _ {n} .y _ {m} ) \rightarrow \infty $.  
 +
Any geodesic ray in $  X $
 +
is convergent, and, conversely, for any class $  \gamma \in \partial  X $
 +
and any point $  x $
 +
there exists a geodesic ray (not necessarily unique!) joining $  x $
 +
and $  \gamma $,  
 +
i.e., starting from $  x $
 +
and belonging to the class $  \gamma $.  
 +
Equivalently, $  \partial  X $
 +
can be defined as the set of asymptotic classes of geodesic rays, with two rays being asymptotic if they are within bounded distance from each other.
  
The hyperbolic boundary is the boundary of the hyperbolic compactification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024049.png" />. For trees the hyperbolic compactification coincides with the end compactification, and for hyperbolic Cartan–Hadamard manifolds with the visibility compactification. The Gromov product naturally extends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024050.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024051.png" />, for sufficiently small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024052.png" />, is uniformly equivalent to a [[Metric|metric]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024053.png" />.
+
The hyperbolic boundary is the boundary of the hyperbolic compactification of $  X $.  
 +
For trees the hyperbolic compactification coincides with the end compactification, and for hyperbolic Cartan–Hadamard manifolds with the visibility compactification. The Gromov product naturally extends to $  \partial  X $,  
 +
and $  e ^ {- \epsilon ( \gamma _ {1} . \gamma _ {2} ) } $,  
 +
for sufficiently small $  \epsilon > 0 $,  
 +
is uniformly equivalent to a [[Metric|metric]] on $  \partial  X $.
  
Hyperbolicity of a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024054.png" /> is determined by its  "global"  structure only. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024056.png" /> are two metric spaces such that there exist mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024058.png" /> and a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024059.png" /> with the property that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024061.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024062.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024063.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024064.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024065.png" /> (such metric spaces are called roughly isometric), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024067.png" /> are hyperbolic simultaneously, and the hyperbolic boundaries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024068.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110240/g11024069.png" /> are homeomorphic.
+
Hyperbolicity of a metric space $  X $
 +
is determined by its  "global"  structure only. If $  ( X,d ) $
 +
and $  ( X  ^  \prime  ,d  ^  \prime  ) $
 +
are two metric spaces such that there exist mappings $  f : X \rightarrow {X  ^  \prime  } $
 +
and $  {f  ^  \prime  } : {X  ^  \prime  } \rightarrow X $
 +
and a constant $  C > 0 $
 +
with the property that $  d  ^  \prime  ( f ( x ) ,f ( y ) ) \leq  C d ( x,y ) + C $
 +
and $  d ( f  ^  \prime  ( f ( x ) ) ,x ) \leq  C $
 +
for all $  x,y \in X $,  
 +
and $  d ( f  ^  \prime  ( x  ^  \prime  ) ,f  ^  \prime  ( y  ^  \prime  ) ) \leq  C d  ^  \prime  ( x  ^  \prime  ,y  ^  \prime  ) + C $
 +
and $  d  ^  \prime  ( f ( f  ^  \prime  ( x  ^  \prime  ) ) ,x  ^  \prime  ) \leq  C $
 +
for all $  x  ^  \prime  ,y  ^  \prime  \in X  ^  \prime  $(
 +
such metric spaces are called roughly isometric), then $  X $
 +
and $  X  ^  \prime  $
 +
are hyperbolic simultaneously, and the hyperbolic boundaries $  \partial  X $
 +
and $  \partial  X  ^  \prime  $
 +
are homeomorphic.
  
 
Isometries of hyperbolic spaces can be classified into elliptic, parabolic and hyperbolic ones in the same way as isometries of the usual hyperbolic plane. An important class of hyperbolic spaces is provided by finitely generated groups whose Cayley graphs are Gromov hyperbolic spaces (such groups are called word hyperbolic).
 
Isometries of hyperbolic spaces can be classified into elliptic, parabolic and hyperbolic ones in the same way as isometries of the usual hyperbolic plane. An important class of hyperbolic spaces is provided by finitely generated groups whose Cayley graphs are Gromov hyperbolic spaces (such groups are called word hyperbolic).

Latest revision as of 19:42, 5 June 2020


hyperbolic space in the sense of Gromov

This notion provides a uniform "global" approach to such objects as the hyperbolic plane, simply-connected Riemannian manifolds with pinched negative sectional curvature, $ { \mathop{\rm CAT} } ( - 1 ) $- spaces, and metric trees. Various "hyperbolic properties" introduced earlier (mostly in the context of group theory) [a1], [a2], [a3] were summed up and further developed by M. Gromov in his seminal paper [a4]. More detailed expositions of (some parts of) Gromov's work can be found in [a5] and [a6].

If $ ( X,d ) $ is a metric space, then the Gromov product of two points $ x,y \in X $ with respect to a reference point $ o \in X $ is defined as

$$ ( x.y ) _ {o} = { \frac{1}{2} } [ d ( o,x ) + d ( o,y ) - d ( x,y ) ] $$

(when $ X $ is a tree, this product coincides with the confluent of $ x $ and $ y $, i.e., the length of the common part of the geodesic segments $ [ o,x ] $ and $ [ o,y ] $). The space $ X $ is called $ \delta $- hyperbolic (with a constant $ \delta \geq 0 $) if the Gromov product satisfies the $ \delta $- ultrametric inequality

$$ ( x.y ) _ {o} \geq \min \{ ( x.z ) _ {o} , ( y.z ) _ {o} \} - \delta $$

for any $ 4 $ points $ o,x,y,z \in X $. If $ X $ is $ \delta $- hyperbolic for some $ \delta \geq 0 $, it is called Gromov hyperbolic. Any metric tree is $ 0 $- hyperbolic, and, conversely, any $ 0 $- hyperbolic space isometrically imbeds into a tree. For Cartan–Hadamard manifolds, hyperbolicity is equivalent to the strong visibility property [a7].

Often one adds to the definition of Gromov hyperbolicity the following two requirements: 1) $ X $ is geodesic (cf. also Geodesic manifold), i.e., any two points in $ X $ can be joined by a path (a geodesic) whose length coincides with the distance between these points; and 2) $ X $ is proper, i.e., all metric balls in $ X $ are compact. Then hyperbolicity is equivalent to the following thin triangles condition: there exists a constant $ \delta ^ \prime $ such that for any geodesic triangle in $ X $ one can choose a point on each of the sides of the triangle in such a way that the pairwise distances between these $ 3 $ points are all not greater than $ \delta ^ \prime $.

Yet another definition of hyperbolicity can be given in terms of exponential divergence of geodesic rays in $ X $, cf. [a8].

A sequence $ ( x _ {n} ) $ in $ X $ is called convergent if $ ( x _ {n} .x _ {m} ) \rightarrow \infty $. The hyperbolic boundary $ \partial X $ of a hyperbolic space $ X $ is defined as the set of equivalence classes of convergent sequences with respect to the equivalence relation $ ( x _ {n} ) \sim ( y _ {n} ) \iff ( x _ {n} .y _ {m} ) \rightarrow \infty $. Any geodesic ray in $ X $ is convergent, and, conversely, for any class $ \gamma \in \partial X $ and any point $ x $ there exists a geodesic ray (not necessarily unique!) joining $ x $ and $ \gamma $, i.e., starting from $ x $ and belonging to the class $ \gamma $. Equivalently, $ \partial X $ can be defined as the set of asymptotic classes of geodesic rays, with two rays being asymptotic if they are within bounded distance from each other.

The hyperbolic boundary is the boundary of the hyperbolic compactification of $ X $. For trees the hyperbolic compactification coincides with the end compactification, and for hyperbolic Cartan–Hadamard manifolds with the visibility compactification. The Gromov product naturally extends to $ \partial X $, and $ e ^ {- \epsilon ( \gamma _ {1} . \gamma _ {2} ) } $, for sufficiently small $ \epsilon > 0 $, is uniformly equivalent to a metric on $ \partial X $.

Hyperbolicity of a metric space $ X $ is determined by its "global" structure only. If $ ( X,d ) $ and $ ( X ^ \prime ,d ^ \prime ) $ are two metric spaces such that there exist mappings $ f : X \rightarrow {X ^ \prime } $ and $ {f ^ \prime } : {X ^ \prime } \rightarrow X $ and a constant $ C > 0 $ with the property that $ d ^ \prime ( f ( x ) ,f ( y ) ) \leq C d ( x,y ) + C $ and $ d ( f ^ \prime ( f ( x ) ) ,x ) \leq C $ for all $ x,y \in X $, and $ d ( f ^ \prime ( x ^ \prime ) ,f ^ \prime ( y ^ \prime ) ) \leq C d ^ \prime ( x ^ \prime ,y ^ \prime ) + C $ and $ d ^ \prime ( f ( f ^ \prime ( x ^ \prime ) ) ,x ^ \prime ) \leq C $ for all $ x ^ \prime ,y ^ \prime \in X ^ \prime $( such metric spaces are called roughly isometric), then $ X $ and $ X ^ \prime $ are hyperbolic simultaneously, and the hyperbolic boundaries $ \partial X $ and $ \partial X ^ \prime $ are homeomorphic.

Isometries of hyperbolic spaces can be classified into elliptic, parabolic and hyperbolic ones in the same way as isometries of the usual hyperbolic plane. An important class of hyperbolic spaces is provided by finitely generated groups whose Cayley graphs are Gromov hyperbolic spaces (such groups are called word hyperbolic).

See also Hyperbolic group.

References

[a1] E. Rips, "Subgroups of small cancellation groups" Bull. London Math. Soc. , 14 (1982) pp. 45–47
[a2] M. Gromov, "Infinite groups as geometric objects" , Proc. Int. Congress Math. Warszawa, 1983 , 1 (1984) pp. 385–391
[a3] J. Cannon, "The combinatorial structure of cocompact discrete hyperbolic groups" Geom. Dedicata , 16 (1984) pp. 123–148
[a4] M. Gromov, "Hyperbolic groups" S.M. Gersten (ed.) , Essays in Group Theory , MSRI Publ. , 8 , Springer (1987) pp. 75–263
[a5] "Sur les groupes hyperboliques d'après Mikhael Gromov" E. Ghys (ed.) P. de la Harpe (ed.) , Progress in Maths. , 83 , Birkhäuser (1990)
[a6] M. Coornaert, T. Delzant, A. Papadopoulos, "Géométrie et théorie des groupes" , Lecture Notes in Mathematics , 1441 , Springer (1990)
[a7] V.A. Kaimanovich, "Ergodicity of harmonic invariant measures for the geodesic flow on hyperbolic spaces" J. Reine Angew. Math. , 455 (1994) pp. 57–103
[a8] J. Cannon, "The theory of negatively curved spaces and groups" T. Bedford (ed.) M. Keane (ed.) C. Ser. (ed.) , Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces , Oxford Univ. Press (1991) pp. 315–369
How to Cite This Entry:
Gromov hyperbolic space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gromov_hyperbolic_space&oldid=17648
This article was adapted from an original article by V.A. Kaimanovich (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article