Classification of isometries and semigroups
6.2. CLASSIFICATION OF SEMIGROUPS
(ii) Letξ be the neutral fixed point ofg. Since Isom(X) acts transitively on
∂X (Proposition 2.5.9), we may conjugate toEin a way suchg(∞) =∞. Then by Proposition 2.5.8 and Example 4.2.11,g is of the form (ii). (iii) Since Isom(X) acts doubly transitively on ∂X (Proposition 2.5.9), we
may conjugate to E in a way such that g+ = 0and g− =∞. Then by
Proposition 2.5.8 and Example 4.2.11, g is of the form (iii). (We have p=0since0∈Fix(g).)
Remark 6.1.12. If g ∈ Isom(X) is elliptic or loxodromic, then the orbit (gn(o))∞
1 exhibits some “regularity” - either it remains bounded forever, or it di-
verges to the boundary. On the other hand, if g is parabolic then the orbit can oscillate, both accumulating at infinity and returning infinitely often to a bounded region. This is in sharp contrast to finite dimensions, where such behavior is im- possible. We discuss such examples in detail in§11.1.2.
6.2. Classification of semigroups
Notation 6.2.1. We denote the set ofglobal fixed points of a semigroupG
Isom(X) by
Fix(G) := \ g∈G
Fix(g). Definition 6.2.2. Gis
• elliptic ifG(o) is a bounded set.
• parabolic ifGis not elliptic and has a global fixed pointξ∈Fix(G) such that
g′(ξ) = 1
∀g∈G,
i.e. ξis neutral with respect to every element ofG.
• loxodromic if it contains a loxodromic isometry. Below we shall prove the following theorem:
Theorem6.2.3. Every semigroup of isometries of a hyperbolic metric space is either elliptic, parabolic, or loxodromic.
Observation6.2.4. An isometrygis elliptic, parabolic, or loxodromic accord- ing to whether the cyclic group generated by it is elliptic, parabolic, or loxodromic. A similar statement holds if “group” is replaced by “semigroup”. Thus, Theorem 6.1.4 is a special case of Theorem 6.2.3.
Before proving Theorem 6.2.3, let us say a bit about each of the different categories in this classification.
6.2.1. Elliptic semigroups. Elliptic semigroups are the least interesting of the semigroups we consider. Indeed, we observe that any strongly discrete ellip- tic semigroup is finite. We now consider the question of whether every elliptic semigroup has a global fixed point.
Theorem 6.2.5 (Cartan’s lemma). If X is a CAT(0) space (and in particular if X is a CAT(-1) space), then every elliptic subsemigroup G Isom(X) has a global fixed point.
We remark that ifGis a group, then this result may be found as [39, Corollary II.2.8(1)].
Proof. SinceG(o) is a bounded set, it has a unique circumcenter [39, Propo- sition II.2.7], i.e. the minimum
min
x∈Xgsup∈Gd(x, g(o))
is achieved at a single pointx∈X. We claim thatxis a global fixed point ofG. Indeed, for eachh∈Gwe have
sup g∈G d(h−1(x), g(o)) = sup g∈G d(x, hg(o))≤sup g∈G d(x, g(o));
sincexis the circumcenter we deduce thath−1(x) =x, or equivalently thath(x) =
x.
On the other hand, if we do not restrict to CAT(0) spaces, then it is possible to have an elliptic group with no global fixed point. We have the following simple example:
Example 6.2.6. Let X = B\BB(0,1) and let g(x) = −x. Then X is a hyperbolic metric space,gis an isometry ofX, andG={id, g}is an elliptic group with no global fixed point.
6.2.2. Parabolic semigroups. Parabolic semigroups will be important in Chapter 12 when we consider geometrically finite semigroups. In particular, we make the following definition:
Definition6.2.7. LetGIsom(X). A pointξ∈∂X is aparabolic fixed point ofGif the semigroup
Gξ:= Stab(G;ξ) ={g∈G:g(ξ) =ξ} is a parabolic semigroup.
In particular, ifGis a parabolic semigroup then the unique global fixed point ofGis a parabolic fixed point.
6.2. CLASSIFICATION OF SEMIGROUPS 101
Warning 6.2.8. A parabolic group does not necessarily contain a parabolic isometry; see Example 11.2.18.
Note that Proposition 4.2.8 yields the following observation:
Observation 6.2.9. LetGIsom(X), and let ξbe a parabolic fixed point of
G. Then the action ofGξ on (Eξ, Dξ) is uniformly Lipschitz, i.e.
Dξ(g(y1), g(y2))≍×Dξ(y1, y2) ∀y1, y2∈ Eξ ∀g∈G,
and the implied constant is independent ofg ∈G. Furthermore, ifX is strongly hyperbolic, thenGacts isometrically onEξ.
Observation 6.2.10. LetGIsom(X), and letξ be a parabolic fixed point ofG. Then for allg∈Gξ,
(6.2.1) Dξ(o, g(o))≍×b(1/2)kgk,
with equality ifX is strongly hyperbolic.
Proof. This is a direct consequence of (3.6.6), (h) of Proposition 3.3.3, and
Proposition 4.2.16.
As a corollary we have the following:
Observation 6.2.11. LetGIsom(X), and letξ be a parabolic fixed point ofG. Then for any sequence (gn)∞1 in Gξ,
kgnk −→ n ∞ ⇔gn(o)−→n ξ. Proof. Indeed, gn(o)−→ n ξ⇔Dξ(o, gn(o))−→n ∞ ⇔ kgnk −→n ∞. Remark6.2.12. IfX is anR-tree, then any parabolic group must be infinitely generated. This follows from a straightforward modification of the proof of Remark 6.1.8.
6.2.3. Loxodromic semigroups. We now come to loxodromic semigroups, which are the most diverse out of these classes. In fact, they are so diverse that we separate them into three subclasses.
Definition6.2.13 ([48]). LetGIsom(X) be a loxodromic semigroup. Gis
• lineal if Fix(g) = Fix(h) for all loxodromicg, h∈G.
• of general type if it has two loxodromic elementsg, h∈Gwith Fix(g)∩Fix(h) =.
• focal if #(Fix(G)) = 1.
(We remark that focal groups were calledquasiparabolic by Gromov [85, §3, Case 4’].
We observe that any cyclic loxodromic group or semigroup is lineal, so this refined classification does not give any additional information for individual isome- tries.
Proposition 6.2.14. Any loxodromic semigroup is either lineal, focal, or of general type.
Proof. Clearly, #(Fix(G))≤ 2 for any loxodromic semigroup G; moreover, #(Fix(G)) = 2 if and only if G is lineal. So to complete the proof, it suffices to show that #(Fix(G)) = 0 if and only if G is of general type. The backward direction is obvious. Suppose that #(Fix(G)) = 0, but thatGis not of general type. Combinatorial considerations show that there exist three pointsξ1, ξ2, ξ3∈∂Xsuch
that Fix(g)⊆ {ξ1, ξ2, ξ3} for allg∈G. But then the set{ξ1, ξ2, ξ3}would have to
be preserved by every element ofg, which contradicts the definition of a loxodromic
isometry.
Let G be a focal semigroup, and let ξ be the global fixed point of G. The dynamics of G will be different depending on whether or not g′(ξ) > 1 for any
g∈G.
Definition6.2.15. Gwill be calledoutward focal ifg′(ξ)>1 for someg
∈G, andinward focal otherwise.
Note that an inward focal semigroup cannot be a group.
Proposition6.2.16. For G≤Isom(X), the following are equivalent: (A) Gis focal.
(B) Ghas a unique global fixed pointξ∈∂X, and g′(ξ)
6
= 1 for someg∈G. (C) G has a unique global fixed pont ξ ∈ ∂X, and there are two loxodromic
isometries g, h∈Gso that g+=h+=ξ, butg− 6=h−.
Proof. The implications (C)⇒(A)⇔(B) are straightforward. Suppose that
G is focal, and let g ∈G be a loxodromic isometry. SinceG is a group, we may without loss of generality suppose that g′(ξ) <1, so that g
+ =ξ. Let j ∈ G be
such that g− ∈/ Fix(j). By choosing n sufficiently large, we may guarantee that
(jgn)′(ξ)<1. Then ifh=jgn, thenhis loxodromic andh
+=ξ. Butg−∈/ Fix(h),