are continuous.
Proof. Fixg0, h0 ∈Isom(X), and let G({x}, U) be a neighborhood ofg0h0.
For someε >0, we haveB(g0h0(x), ε)⊆U. We claim that G {h0(x)}, B(g0h0(x), ε/2) G {x}, B(h0(x), ε/2) ⊆G({x}, U).
Indeed, fixg∈G({h0(x)}, B(g0h0(x), ε/2)) andh∈G({x}, B(h0(x), ε/2)). Then d(gh(x), g0h0(x))≤d(h(x), h0(x)) +d(gh0(x), g0h0(x))≤ε/2 +ε/2 =ε,
demonstrating thatgh(x)∈U, and thus that the map (g, h)7→ghis continuous. Now fixg0∈Isom(X), and letG({x}, U) be a neighborhood ofg0−1. For some ε >0, we haveB(g0−1(x), ε)⊆U. We claim that
G {g0−1(x)}, B(x, ε) −1
⊆G({x}, U).
Indeed, fixg∈G({g0−1(x)}, B(x, ε)). Then d(g−1(x), g−1
0 (x)) =d(x, gg
−1
0 (x))≤ε,
demonstrating thatg−1(x)∈U, and thus that the mapg7→g−1is continuous. Remark5.1.4 ([109, 9.B(9), p.60]). IfX is a separable complete metric space, then the group Isom(X) with the compact-open topology is a Polish space.
5.2. Discrete groups of isometries
In this section we discuss several different notions of what it means for a group
G ≤ Isom(X) to be discrete, and then we show that they are equivalent in the Standard Case. However, each of our notions will be distinct whenX =H=HαF for some infinite cardinalα.
Definition 5.2.1. FixG≤Isom(X).
• Gis calledstrongly discrete (SD)if for every bounded setB⊆X, we have #{g∈G:g(B)∩B6=}<∞.
• Gis called moderately discrete (MD) if for every x∈X, there exists an open setU ∋xsuch that
#{g∈G:g(U)∩U 6=}<∞.
• Gis calledweakly discrete (WD) if for everyx∈X, there exists an open setU ∋xsuch that
Remark 5.2.2. Strongly discrete groups are known in the literature asmetri- cally proper, and moderately discrete groups are known as wandering.
Remark 5.2.3. We may equivalently give the definitions as follows:
• Gisstrongly discrete (SD)if for everyR >0 andx∈X, (5.2.1) #{g∈G:d(x, g(x))≤R}<∞.
• Gismoderately discrete (MD)if for everyx∈X, there existsε >0 such that
(5.2.2) #{g∈G:d(x, g(x))≤ε}<∞.
• Gisweakly discrete (WD)if for everyx∈X, there existsε >0 such that (5.2.3) G(x)∩B(x, ε) ={x}.
As our naming suggests, the condition of strong discreteness is stronger than the condition of moderate discreteness, which is in turn stronger than the condition of weak discreteness.
Proposition 5.2.4. Any strongly discrete group is moderately discrete, and any moderately discrete group is weakly discrete.
Proof. It is clear from the second formulation that strongly discrete groups are moderately discrete. Let G ≤Isom(X) be a moderately discrete group. Fix
x∈X, and let ε >0 be such that (5.2.2) holds. Lettingε′ =ε∧min{d(x, g(x)) :
g(x)6=x, g(x)∈B(x, ε)}, we see that (5.2.3) holds. The reverse directions, WD⇒MD and MD⇒SD, both fail in infinite dimen- sions. Examples 11.1.14 and 13.3.1-13.3.3 are moderately discrete groups which are not strongly discrete, and Examples 13.5.2 and 13.4.1 are weakly discrete groups which are not moderately discrete.
IfXis a proper metric space, then the classes MD and SD coincide, but are still distinct from WD. Example 13.4.1 is a weakly discrete group acting on a proper metric space which is not moderately discrete. We show now that MD⇔SD when
X is proper:
Proposition5.2.5. Suppose thatX is proper. Then a subgroup ofIsom(X)is moderately discrete if and only if it is strongly discrete.
Proof. LetG≤Isom(X) be a moderately discrete subgroup. Fixx∈X, and let ε > 0 satisfy (5.2.2). Fix R > 0 and let K = G(o)∩B(x, R); K is compact sinceX is proper. The collection{B(g(x), ε) :g∈G} coversK, so there is a finite
5.2. DISCRETE GROUPS OF ISOMETRIES 89 subcover{B(gi(x), ε) :i= 1, . . . , n}. Now #{g∈G:d(x, g(x)≤R)} ≤ n X i=1 #{g∈G:g(x)∈B(gi(x), ε)}<∞, i.e. (5.2.1) holds. 5.2.1. Topological discreteness.
Definition 5.2.6. LetT be a topology on Isom(X). A groupG≤Isom(X)
isT-discrete if it is discrete as a subspace of Isom(X) in the topologyT.
Most of the time, we will let T be the compact-open topology (COT). The
relation between COT-discreteness and our previous notions of discreteness is as follows:
Proposition5.2.7.
(i) Any moderately discrete group is COT-discrete.
(ii) Any weakly discrete group that is acting on an algebraic hyperbolic space is COT-discrete.
(iii) Any COT-discrete group that is acting on a proper metric space is strongly discrete.
Proof.
(i) Let G≤ Isom(X) be moderately discrete, and let ε > 0 satisfy (5.2.2). Then the set U := G({o}, B(o, ε)) ⊆ Isom(X) satisfies #(U ∩G) < ∞. ButU is a neighborhood of id in the compact-open topology. It follows thatGis COT-discrete.
(ii) Suppose thatX =H=HαF. LetG≤Isom(H) be weakly discrete, and by contradiction suppose it is not COT-discrete. For any finite set F ⊆H, letε >0 be small enough so that (5.2.3) holds for all x∈ F; since Gis not COTD, there existsg=gF ∈G\ {id}such thatd(x, g(x))≤εfor all
x∈F, and it follows thatg(x) =xfor allx∈F. Now suppose thatJ is a finite set of indices, and letF ={[e0]} ∪ {[e0±(1/2)ei]ℓ:i∈J, ℓ∈IF}, where IF is as in (5.1.2). Then if TI is a representative of gF satisfying
TJe0=e0, an argument similar to the proof of Proposition 5.1.2(ii) shows
thatσTJ =Iand TJei=ei for alli∈J.
Now we define an infinite sequence of indices (in)∞1 as follows: If i1, . . . , in−1 have been defined, let Tn = T{i1,...,in−1}, and let in be such
thatein ∈/ Fix(Tn).
Choose a nonnegative summable sequence (tn)∞1 , and let x = e0+ P∞
Tnx=xfor allnsufficiently large. Fix such ann, and observe that 0 =Tnx−x=tn(Tn(en)−en) +
X
m>n
tm(Tn(em)−em); the triangle inequality gives
tn≤
P
m>n2tm
kTnen−enk· By choosing the sequence (tn)∞1 to satisfy
tn+1< 1 4kTnen−enktn≤ 1 2tn, we arrive at a contradiction.
(iii) Let G be a COT-discrete group acting by isometries on a proper metric spaceX. By contradiction, suppose thatGis not strongly discrete. Then there exists an infinite set A ⊆ G such that the set A(o) is bounded. Without loss of generality we may suppose thatA−1=A. Note that for
each x ∈ X, the set A(x) is bounded and therefore precompact. Now sinceX is a proper metric space, it isσ-compact and therefore separable. LetS be a countable dense subset ofX. Then
K:= Y q∈S A(q) 2
is a compact metrizable space. For eachg∈A let
φg:= (g(q))q∈S,(g−1(q))q∈S∈ K.
SinceAis infinite, there exists an infinite sequence (gn)∞1 in Asuch that φgn→ (yq(+))q∈S,(y(q−))q∈S ∈ K. Thus gn±(q)−→ n y (±) q ∀q∈ S.
The density of S and the equicontinuity of the sequences (gn)∞1 and
(g−1
n )∞1 imply that for allx∈X, there existy (±)
x such thatgn±(y)→y
(±)
x . Thus, the sequence (gn)∞1 converges in the Tychonoff topology to some g(+)
∈XX. Similarly, the sequence (g−1
n )∞1 converges to someg(−)∈XX.
Again, the equicontinuity of the sequences (gn)∞1 and (gn−1)∞1 allows us
to take limits and deduce that
g(+)g(−)= lim n→∞gng
−1
5.2. DISCRETE GROUPS OF ISOMETRIES 91
Similarly,g(−)g(+)= id. Thusg(+)andg(−)are inverses, and in particular g(+)
∈Isom(X). Sincegn→g(+)in the compact-open topology, the proof is completed by the following lemma from topological group theory:
Lemma 5.2.8. Let H be a topological group, and let Gbe a subgroup of H. Suppose there is a sequence (gn)∞1 of distinct elements inGwhich
converges to an element of H. Then G is not discrete in the topology inherited from H.
Proof. Supposegn→h∈H. Then
gngn−+11 →hh
−1= id,
while on the other hand gngn−+11 6= id (since the sequence (gn)∞1 consists
of distinct elements). This demonstrates that G is not discrete in the
inherited topology. ⊳
IfXis not an algebraic hyperbolic space, then it is possible for a weakly discrete group to not be COT-discrete; see Example 13.4.1. Conversely, it is possible for a COT-discrete group to not be weakly discrete; see Examples 13.4.9 amd 13.5.1.
On the other hand, suppose that X is an algebraic hyperbolic space. The uniform operator topology (abbreviated as UOT) is finer than the COT, i.e. it has more open sets, and therefore it is easier for every subset ofGto be relatively open in that topology, which is exactly what it means to be discrete. Notice that there is an “order switch” here; the UOT is finer than the COT, but the condition of being COT-discrete is stronger than the condition of being UOT-discrete. We record this for later use as the following
Observation 5.2.9. Let X be an algebraic hyperbolic space. If a subgroup
G≤Isom(X) is COT-discrete, then it is also UOT-discrete.
The inclusion in the previous observation is strict. A significant example of a group acting onH∞ which is UOT-discrete but not COT-discrete is described in Example 13.4.2.
The various relations between the distinct shades of discreteness are somewhat subtle when first discerned. We speculate that it may be fruitful to study such distinctions with a finer lens. For the reader’s ease, we summarize the relations between our different notions of discreteness in Table 1 below.
Proposition5.2.10. Suppose thatX is a finite-dimensional Riemannian man- ifold. Then the notions of strong discreteness, moderate discreteness, weak dis- creteness, and COT-discreteness agree. IfX is an algebraic hyperbolic space, these notions also agree with the notion of UOT-discreteness.
Proof. By Propositions 5.2.4 and 5.2.7, the conditions of strong discreteness, moderate discreteness, and COT-discreteness agree and imply weak discreteness. Conversely, suppose that G ≤ Isom(X) is weakly discrete, and by contradiction suppose thatGis not COT-discrete. SinceX is separable, so is Isom(X), and thus there exists a sequence Isom(X)\{id} ∋gn→id in the compact-open topology. For eachnletFn={x∈X :gn(x) =x}. SinceGis weakly discrete,X =S∞1 Fn, so by the Baire category theorem,Fnhas nonempty interior for somen. But thengn= id on an open set; in particular there exists a point x0 ∈ X such that gn(x0) = x0
andg′
n(x0) is the identity map on the tangent space ofx0. By the naturality of the
exponential map, this implies thatgn is the identity map, a contradiction.
Finally, suppose X =H =HαF is an algebraic hyperbolic space, and let L =
LαF+1. Since L is finite-dimensional, the SOT and UOT topologies on L(L) are equivalent. This in turn demonstrates that the notions of COT-discreteness and
UOT-discreteness agree.
In such a setting, we shall call a group satisfying any of these equivalent defi- nitions simplydiscrete.
5.2.3. Proper discontinuity.
Definition5.2.11. A groupG≤Isom(X)acts properly discontinuously (PrD) onX if for everyx∈X, there exists an open setU∋xwith
g(U)∩U 6=⇒g= id,
or equivalently, if
d(x,{g(x) :g6= id})>0.
Let us discuss the relations between proper discontinuity and some of our no- tions of discreteness. We begin by noting that even in finite dimensions, the notion of proper discontinuity is not the same as the notion of discreteness; instead, a group acts properly discontinuously if and only if it both discrete and torsion-free. We also remark that in finite dimensions Selberg’s lemma (see e.g. [8]) can be used to pass from a discrete group to a finite-index subgroup that acts properly discon- tinuously. However, it is impossible to do this in infinite dimensions; cf. Example 11.2.18.
Although no notion of discreteness implies proper discontinuity, the reverse is true for certain types of discreteness. Namely, since #{id}= 1<∞, we have:
5.2. DISCRETE GROUPS OF ISOMETRIES 93
Observation 5.2.12. Any group which acts properly discontinuously is mod- erately discrete.
In particular, by combining with Proposition 5.2.5 we see that ifX is proper then any group which acts properly discontinuously is strongly discrete. This pro- vides a connection between our results, in which strong discreteness is often a hypothesis, and many results from the literature in which proper discontinuity and properness are both hypotheses.
Observation 5.2.12 admits the following partial converse, which generalizes the fact that in finite dimensions every discrete torsion-free group acts properly discon- tinuously:
Remark 5.2.13. If X is a proper CAT(0) space, then a group acts properly discontinously if and only if it is moderately discrete and torsion free.
Proof. Suppose that G ≤ Isom(X) acts properly discontinuously. If g ∈ G\ {id}is a torsion element, then by Cartan’s lemma [39, II.2.8(1)],g has a fixed point. This contradictsGacting properly discontinuously. ThusGis torsion-free.
Conversely, suppose thatG≤Isom(X) is moderately discrete and torsion-free. Given x∈X, let ε >0 be as in (5.2.3), and by contradiction suppose that there exists g6= id such that d(x, g(x))< ε. By (5.2.3),g(x) = x. But then by (5.2.2), the set{gn:n∈Z}is finite, i.e. g is a torsion element. This is a contradiction, so
Gacts properly discontinuously.
We summarize the relations between our various notions of discreteness, to- gether with proper discontinuity, in the following table:
Finite dimensional SD ↔ MD ↔ WD
Riemannian manifold ↑ l
PrD COTD ↔ UOTD
SD → MD → WD
General metric space ր ց
PrD COTD
Infinite dimensional SD → MD → WD algebraic hyperbolic space ր ↓
PrD COTD → UOTD
SD ↔ MD ↔ COTD
Proper metric space ↑ ↓
PrD WD
Table 1. The relations between different notions of discreteness. COTD and UOTD stand for discrete with respect to the compact- open and uniform operator topologies respectively. All implica- tions not listed have counterexamples; see Chapter 13.
5.2.4. Behavior with respect to restrictions. Fix G ≤ Isom(X), and suppose Y ⊆X is a subspace of X preserved by G, i.e. g(Y) =Y for all g ∈G. ThenGcan be viewed as a group acting on the metric space (Y, d↿Y).
Observation 5.2.14.
(i) Gis strongly discrete⇔G↿Y is strongly discrete (ii) Gis moderately discrete⇒G↿Y is moderately discrete (iii) Gis weakly discrete⇒G↿Y is weakly discrete
(iv) GisT-discrete⇐G↿Y isT ↿Y-discrete
(v) Gacts properly discontinuously onX ⇒Gacts properly discontinuously onY.
In particular, strong discreteness is the only concept which is “independent of the space being acted on”. It is thus the most robust of all our definitions.
Note that for the notions of topological discreteness like COTD and UOTD, the order of implication reverses; restricting to a subspace may cause a group to no longer be discrete. Example 13.4.9 is an example of this phenomenon.
5.2.5. Countability of discrete groups. In finite dimensions, all discrete groups are countable. In general, it depends on what type of discreteness you are considering.
Proposition5.2.15. Fix G≤Isom(X), and suppose that either (1) Gis strongly discrete, or
(2) X is separable and Gis COT-discrete. Then Gis countable.
Proof. IfGis strongly discrete, then #(G)≤X
n∈N
#{g∈G:kgk ≤n} ≤ X
n∈N
#(N) = #(N).
On the other hand, ifXis a separable metric space, then by Remark 5.1.4 Isom(X) is separable metrizable, so it contains no uncountable discrete subspaces. Remark 5.2.16. An example of an uncountable UOT-discrete subgroup of Isom(H∞) is given in Example 13.4.2, and an example of an uncountable weakly
discrete group acting on a separableR-tree is given in Example 13.4.1. An example of an uncountable moderately discrete group acting on a (non-separable)R-tree is given in Remark 13.3.4.
CHAPTER 6