On the properly discontinuous groups of isometries of hyperbolic spaces in the sense of Gromov.
(Sur les groupes proprement discontinus d‘isométries des espaces hyperboliques au sens de Gromov.)

*(French)*Zbl 0777.53044
Publication de l’Institut de Recherche Mathématique Avancée. 444. Strasbourg: Université Louis Pasteur, Departement de Mathématique, xvi, 124 p. (1990).

This article, which represents the doctoral dissertation of the author, extends to hyperbolic spaces in the sense of Gromov several geometric features of the real hyperbolic spaces with constant negative sectional curvature. The article consists of three parts. The first is a lucid summary, sometimes with proofs, of the basic features of hyperbolic spaces. [See also E. Ghys and P. de la Harpe, Sur les groupes hyperboliques d’après Mikhail Gromov (1990; Zbl 0731.20025) and M. Gromov, Hyperbolic groups, in: Essays in group theory, Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)]. In the first part the author also states the results of the last two parts, which are self contained articles on domains of discontinuity and Patterson-Sullivan measures respectively.

Let \(X\) be a metric space with distinguished base point \(x_ 0\). For points \(x,y\) of \(X\) one defines the Gromov product \((x\cdot y)={1\over 2}\{d(x,x_ 0)+d(y,x_ 0)-d(x,y)\}\). The space \(X\) is said to be \(\delta\)-hyperbolic for some constant \(\delta\geq 0\) if \((x\cdot y)\geq\min\{(x\cdot z),(y\cdot z)\}-\delta\) for all points \(x,y,z\) of \(X\). A finitely generated group \(\Gamma\) is said to be \(\delta\)-hyperbolic with respect to a finite generating set \(S\) if the Cayley graph of \(\Gamma\) equipped with the word metric from \(S\) is \(\delta\)-hyperbolic. If \(\Gamma\) is \(\delta\)-hyperbolic for some \(\delta>0\) and some finite generating set \(S\), then for any other finite generating set \(S'\) there exists \(\delta'>0\) such that \(\Gamma\) is \(\delta'\)-hyperbolic. A metric space \(X\) is a geodesic space if any two points of \(X\) can be joined by a geodesic, which is a curve \(\sigma:I\to X\) such that \(d(\sigma s,\sigma t)=| s-t|\) for all \(s,t\) in the interval \(I\). For geodesic spaces \(X\) the condition of \(\delta\)-hyperbolicity is essentially equivalent to the following \(\delta\)-condition of I. Rips: Let \(\Delta\) be a geodesic triangle in \(X\). Then every point of \(\Delta\) lies at distance at most \(\delta\) from the union of the two sides of \(\Delta\) that do not contain \(x\). Examples of spaces or groups that are \(\delta\)-hyperbolic include the following: 1) Trees \((\delta=0)\). 2) Complete, simply connected Riemannian manifolds \(X\) with sectional curvature bounded above by a negative constant \((\delta>0)\). 3) A cocompact lattice \(\Gamma\) of isometries in a space \(X\) as in 2) above \((\delta>0)\).

A metric space \(X\) is a tree if any subset of \(X\) that is homeomorphic to an interval \(I\) of \(\mathbb{R}\) is actually the image of a geodesic segment. Gromov has shown that any \(\delta\)-hyperbolic space can be approximated arbitrarily closely by a tree, and this fact plays an important role in the proofs of the author in this article.

Many geometric properties that are well known for complete, simply connected manifolds with sectional curvature bounded above by a negative constant can be extended to \(\delta\)-hyperbolic spaces \(X\). In part \(A\) of the article the author describes many such properties. In particular a \(\delta\)-hyperbolic space \(X\) admits a boundary \(\partial X\) that equals the boundary sphere of equivalence classes of asymptotic geodesics if \(X\) is smooth with negative sectional curvature. For each choice of a base point \(x_ 0\) in \(X\) and each number \(a>1\), the boundary \(\partial X\) admits a natural “visual” metric. Any two points \(x,y\) of \(\partial X\) may be joined by a geodesic \(\sigma\), and if the distance between \(x_ 0\) and \(\sigma\) is large, then the distance in \(\partial X\) between \(x\) and \(y\) is small with respect to the visual metric. One may define Busemann functions, Poisson kernels and a notion of quasiconformality for the action of isometries of \(X\) on \(\partial X\). If \(\Gamma\) is a group of isometries of a \(\delta\)-hyperbolic space \(X\), then \(\Gamma\) determines a limit set \(L(\Gamma)\subseteq\partial X\), which is invariant under \(\Gamma\). The topological action of \(\Gamma\) on \(L(\Gamma)\) or on \(L(\Gamma)\times L(\Gamma)\), or more generally the topological action of an isometry of \(X\) on \(\partial X\), is the same as for smooth manifolds with strictly negative curvature. The Patterson-Sullivan notion of a \(\Gamma\)-conformal measure of dimension \(D\) on \(\partial H^ n\) extends to the notion of \(\Gamma\)-quasi conformal measure of dimension \(D\) on \(\partial X\).

In part B the author proves that if \(\Gamma\) is a properly discontinuous group of isometries of a \(\delta\)-hyperbolic space \(X\) for some \(\delta\geq 0\), then \(\Gamma\) acts properly discontinuously on \(X\cup\Omega(\Gamma)\), where \(\Omega(\Gamma)=\partial X-L(\Gamma)\). In part C the theory of Patterson-Sullivan measures on \(\partial H^ n\) is extended to \(\partial X\), where \(X\) is a \(\delta\)-hyperbolic space for some \(\delta\geq 0\). In such a space we fixe a base point \(x_ 0\) in \(X\) and a number \(a>1\), which determine a visual metric in \(\partial X\) and corresponding notions of quasiconformality for functions and measures on \(\partial X\). A group \(\Gamma\) of isometries of \(X\) is said to be quasi convex cocompact if it acts properly discontinuously on \(X\) and if \(Q(\Gamma)/\Gamma\) is compact, where \(Q(\Gamma)\) denotes the union of all geodesics in \(X\) that join points of \(L(\Gamma)\subseteq\partial X\). Let \(\Gamma\) be a quasi convex cocompact group of isometries of \(X\), and let \(Y\) be an orbit of \(\Gamma\) in \(X\). For \(R>0\) define \(n_ Y(R)\) to be the number of points of \(Y\) that lie inside \(B(x_ 0,R)\), the open ball of radius \(R\) with center \(x_ 0\). Define the critical exponent \(e_ a(\Gamma)=\limsup_{R\to\infty}(1/R)\log_ an_ Y(R)\). (If \(a=e\), then we write \(e(\Gamma)\) instead of \(e_ a(\Gamma))\). The critical exponent depends neither on \(x_ 0\) nor on \(Y\). The author obtains the following results, which were proved in sometimes sharper form by Patterson and Sullivan for real hyperbolic spaces \(H^ n\):

Theorem. Let \(X\) be a \(\delta\)-hyperbolic space and let \(\Gamma\) be a quasi convex cocompact group of isometries of \(X\). Assume that the critical exponent \(e_ a(\Gamma)\) is finite. Then 1) If \({\mathcal H}\) denotes the Hausdorff measure on \(\partial X\) of dimension \(D=e_ a(\Gamma)\), then \({\mathcal H}\) has support in \(L(\Gamma)\) and is \(\Gamma\)- quasi conformal on \(\partial X\). Moreover \({\mathcal H}(L(\Gamma))\) is finite and nonzero. 2) If \(\mu\) is any \(\Gamma\)-quasi conformal measure on \(\partial X\) of dimension \(D\geq 0\) whose support lies in \(L(\Gamma)\), then \(D=e_ a(\Gamma)\) and \(\mu\) is absolutely equivalent to \({\mathcal H}\). Moreover, the Radon-Nikodym derivative \(d\mu/d{\mathcal H}\) is uniformly bounded above and below in absolute value by positive constants. 3) The action of \(\Gamma\) on \((L(\Gamma),{\mathcal H})\) is ergodic. 4) Let \(Y\) be an orbit of \(\Gamma\) in \(X\). Then a) There exists a real number \(C\geq 1\) such that \(C^{-1}\exp(e(\Gamma)R)\leq n_ Y(R)\leq C\exp(e(\Gamma)R)\) for all \(R>0\). b) The Poincaré series \(g_ Y(s)=\sum_{y\in Y}a^{- sd(y,x_ 0)}\) diverges for \(s<e_ a(\Gamma)\) and converges for \(s>e_ a(\Gamma)\).

Let \(X\) be a metric space with distinguished base point \(x_ 0\). For points \(x,y\) of \(X\) one defines the Gromov product \((x\cdot y)={1\over 2}\{d(x,x_ 0)+d(y,x_ 0)-d(x,y)\}\). The space \(X\) is said to be \(\delta\)-hyperbolic for some constant \(\delta\geq 0\) if \((x\cdot y)\geq\min\{(x\cdot z),(y\cdot z)\}-\delta\) for all points \(x,y,z\) of \(X\). A finitely generated group \(\Gamma\) is said to be \(\delta\)-hyperbolic with respect to a finite generating set \(S\) if the Cayley graph of \(\Gamma\) equipped with the word metric from \(S\) is \(\delta\)-hyperbolic. If \(\Gamma\) is \(\delta\)-hyperbolic for some \(\delta>0\) and some finite generating set \(S\), then for any other finite generating set \(S'\) there exists \(\delta'>0\) such that \(\Gamma\) is \(\delta'\)-hyperbolic. A metric space \(X\) is a geodesic space if any two points of \(X\) can be joined by a geodesic, which is a curve \(\sigma:I\to X\) such that \(d(\sigma s,\sigma t)=| s-t|\) for all \(s,t\) in the interval \(I\). For geodesic spaces \(X\) the condition of \(\delta\)-hyperbolicity is essentially equivalent to the following \(\delta\)-condition of I. Rips: Let \(\Delta\) be a geodesic triangle in \(X\). Then every point of \(\Delta\) lies at distance at most \(\delta\) from the union of the two sides of \(\Delta\) that do not contain \(x\). Examples of spaces or groups that are \(\delta\)-hyperbolic include the following: 1) Trees \((\delta=0)\). 2) Complete, simply connected Riemannian manifolds \(X\) with sectional curvature bounded above by a negative constant \((\delta>0)\). 3) A cocompact lattice \(\Gamma\) of isometries in a space \(X\) as in 2) above \((\delta>0)\).

A metric space \(X\) is a tree if any subset of \(X\) that is homeomorphic to an interval \(I\) of \(\mathbb{R}\) is actually the image of a geodesic segment. Gromov has shown that any \(\delta\)-hyperbolic space can be approximated arbitrarily closely by a tree, and this fact plays an important role in the proofs of the author in this article.

Many geometric properties that are well known for complete, simply connected manifolds with sectional curvature bounded above by a negative constant can be extended to \(\delta\)-hyperbolic spaces \(X\). In part \(A\) of the article the author describes many such properties. In particular a \(\delta\)-hyperbolic space \(X\) admits a boundary \(\partial X\) that equals the boundary sphere of equivalence classes of asymptotic geodesics if \(X\) is smooth with negative sectional curvature. For each choice of a base point \(x_ 0\) in \(X\) and each number \(a>1\), the boundary \(\partial X\) admits a natural “visual” metric. Any two points \(x,y\) of \(\partial X\) may be joined by a geodesic \(\sigma\), and if the distance between \(x_ 0\) and \(\sigma\) is large, then the distance in \(\partial X\) between \(x\) and \(y\) is small with respect to the visual metric. One may define Busemann functions, Poisson kernels and a notion of quasiconformality for the action of isometries of \(X\) on \(\partial X\). If \(\Gamma\) is a group of isometries of a \(\delta\)-hyperbolic space \(X\), then \(\Gamma\) determines a limit set \(L(\Gamma)\subseteq\partial X\), which is invariant under \(\Gamma\). The topological action of \(\Gamma\) on \(L(\Gamma)\) or on \(L(\Gamma)\times L(\Gamma)\), or more generally the topological action of an isometry of \(X\) on \(\partial X\), is the same as for smooth manifolds with strictly negative curvature. The Patterson-Sullivan notion of a \(\Gamma\)-conformal measure of dimension \(D\) on \(\partial H^ n\) extends to the notion of \(\Gamma\)-quasi conformal measure of dimension \(D\) on \(\partial X\).

In part B the author proves that if \(\Gamma\) is a properly discontinuous group of isometries of a \(\delta\)-hyperbolic space \(X\) for some \(\delta\geq 0\), then \(\Gamma\) acts properly discontinuously on \(X\cup\Omega(\Gamma)\), where \(\Omega(\Gamma)=\partial X-L(\Gamma)\). In part C the theory of Patterson-Sullivan measures on \(\partial H^ n\) is extended to \(\partial X\), where \(X\) is a \(\delta\)-hyperbolic space for some \(\delta\geq 0\). In such a space we fixe a base point \(x_ 0\) in \(X\) and a number \(a>1\), which determine a visual metric in \(\partial X\) and corresponding notions of quasiconformality for functions and measures on \(\partial X\). A group \(\Gamma\) of isometries of \(X\) is said to be quasi convex cocompact if it acts properly discontinuously on \(X\) and if \(Q(\Gamma)/\Gamma\) is compact, where \(Q(\Gamma)\) denotes the union of all geodesics in \(X\) that join points of \(L(\Gamma)\subseteq\partial X\). Let \(\Gamma\) be a quasi convex cocompact group of isometries of \(X\), and let \(Y\) be an orbit of \(\Gamma\) in \(X\). For \(R>0\) define \(n_ Y(R)\) to be the number of points of \(Y\) that lie inside \(B(x_ 0,R)\), the open ball of radius \(R\) with center \(x_ 0\). Define the critical exponent \(e_ a(\Gamma)=\limsup_{R\to\infty}(1/R)\log_ an_ Y(R)\). (If \(a=e\), then we write \(e(\Gamma)\) instead of \(e_ a(\Gamma))\). The critical exponent depends neither on \(x_ 0\) nor on \(Y\). The author obtains the following results, which were proved in sometimes sharper form by Patterson and Sullivan for real hyperbolic spaces \(H^ n\):

Theorem. Let \(X\) be a \(\delta\)-hyperbolic space and let \(\Gamma\) be a quasi convex cocompact group of isometries of \(X\). Assume that the critical exponent \(e_ a(\Gamma)\) is finite. Then 1) If \({\mathcal H}\) denotes the Hausdorff measure on \(\partial X\) of dimension \(D=e_ a(\Gamma)\), then \({\mathcal H}\) has support in \(L(\Gamma)\) and is \(\Gamma\)- quasi conformal on \(\partial X\). Moreover \({\mathcal H}(L(\Gamma))\) is finite and nonzero. 2) If \(\mu\) is any \(\Gamma\)-quasi conformal measure on \(\partial X\) of dimension \(D\geq 0\) whose support lies in \(L(\Gamma)\), then \(D=e_ a(\Gamma)\) and \(\mu\) is absolutely equivalent to \({\mathcal H}\). Moreover, the Radon-Nikodym derivative \(d\mu/d{\mathcal H}\) is uniformly bounded above and below in absolute value by positive constants. 3) The action of \(\Gamma\) on \((L(\Gamma),{\mathcal H})\) is ergodic. 4) Let \(Y\) be an orbit of \(\Gamma\) in \(X\). Then a) There exists a real number \(C\geq 1\) such that \(C^{-1}\exp(e(\Gamma)R)\leq n_ Y(R)\leq C\exp(e(\Gamma)R)\) for all \(R>0\). b) The Poincaré series \(g_ Y(s)=\sum_{y\in Y}a^{- sd(y,x_ 0)}\) diverges for \(s<e_ a(\Gamma)\) and converges for \(s>e_ a(\Gamma)\).

Reviewer: P.Eberlein (Chapel Hill)

##### MSC:

53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |

58C35 | Integration on manifolds; measures on manifolds |

53C20 | Global Riemannian geometry, including pinching |