Švarc–Milnor Lemma
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In the mathematical subject of
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 group ...
, the Švarc–Milnor lemma (sometimes also called Milnor–Švarc lemma, with both variants also sometimes spelling Švarc as Schwarz) is a statement which says that a group G, equipped with a "nice"
discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory * Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit * Discrete group, ...
isometric action on a
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
X, is quasi-isometric to X. This result goes back, in different form, before the notion of
quasi-isometry In mathematics, a quasi-isometry is a function between two metric spaces that respects large-scale geometry of these spaces and ignores their small-scale details. Two metric spaces are quasi-isometric if there exists a quasi-isometry between them. ...
was formally introduced, to the work of Albert S. Schwarz (1955) and
John Milnor John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Uni ...
(1968). Pierre de la Harpe called the Švarc–Milnor lemma "the ''fundamental observation in geometric group theory''"Pierre de la Harpe,
Topics in geometric group theory
'. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2000. ; p. 87
because of its importance for the subject. Occasionally the name "fundamental observation in geometric group theory" is now used for this statement, instead of calling it the Švarc–Milnor lemma; see, for example, Theorem 8.2 in the book of Farb and Margalit.


Precise statement

Several minor variations of the statement of the lemma exist in the literature. Here we follow the version given in the book of Bridson and Haefliger (see Proposition 8.19 on p. 140 there).M. R. Bridson and A. Haefliger, ''Metric spaces of non-positive curvature''. Grundlehren der Mathematischen Wissenschaften undamental Principles of Mathematical Sciences vol. 319. Springer-Verlag, Berlin, 1999. Let G be a group acting by isometries on a proper length space X such that the action is properly discontinuous and cocompact. Then the group G is finitely generated and for every finite generating set S of G and every point p\in X the orbit map :f_p:(G,d_S)\to X, \quad g\mapsto gp is a
quasi-isometry In mathematics, a quasi-isometry is a function between two metric spaces that respects large-scale geometry of these spaces and ignores their small-scale details. Two metric spaces are quasi-isometric if there exists a quasi-isometry between them. ...
. Here d_S is the word metric on G corresponding to S. Sometimes a properly discontinuous cocompact isometric action of a group G on a proper geodesic metric space X is called a ''geometric'' action.


Explanation of the terms

Recall that a metric space X is ''proper'' if every closed ball in X is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
. An action of G on X is ''properly discontinuous'' if for every compact K\subseteq X the set :\ is finite. The action of G on X is ''cocompact'' if the quotient space X/G, equipped with the
quotient topology In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient to ...
, is compact. Under the other assumptions of the Švarc–Milnor lemma, the cocompactness condition is equivalent to the existence of a closed ball B in X such that :\bigcup_ gB=X.


Examples of applications of the Švarc–Milnor lemma

For Examples 1 through 5 below see pp. 89–90 in the book of de la Harpe. Example 6 is the starting point of the part of the paper of Richard Schwartz. Richard Schwartz, ''The quasi-isometry classification of rank one lattices'', Publications Mathématiques de l'Institut des Hautes Études Scientifiques, vol. 82, 1995, pp. 133–168 # For every n\ge 1 the group \mathbb Z^n is quasi-isometric to the Euclidean space \mathbb R^n. # If \Sigma is a closed connected oriented surface of negative
Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's ...
then the
fundamental group In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...
\pi_1(\Sigma) is quasi-isometric to the hyperbolic plane \mathbb H^2. # If (M,g) is a closed connected smooth manifold with a smooth
Riemannian metric In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
g then \pi_1(M) is quasi-isometric to (\tilde M, d_), where \tilde M is the
universal cover In topology, a covering or covering projection is a map between topological spaces that, intuitively, locally acts like a projection of multiple copies of a space onto itself. In particular, coverings are special types of local homeomorphism ...
of M, where \tilde g is the pull-back of g to \tilde M, and where d_ is the path metric on \tilde M defined by the Riemannian metric \tilde g . # If G is a connected finite-dimensional
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
equipped with a left-invariant
Riemannian metric In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
and the corresponding path metric, and if \Gamma\le G is a uniform lattice then \Gamma is quasi-isometric to G. # If M is a closed hyperbolic 3-manifold, then \pi_1(M) is quasi-isometric to \mathbb H^3. # If M is a complete finite volume hyperbolic 3-manifold with cusps, then \Gamma=\pi_1(M) is quasi-isometric to \Omega= \mathbb H^3-\mathcal B, where \mathcal B is a certain \Gamma-invariant collection of horoballs, and where \Omega is equipped with the induced path metric.


References

{{DEFAULTSORT:Svarc-Milnor lemma Geometric group theory Metric geometry