Journal of Lie Theory Volume 13 (2003) 7–12 ◯^{C}2003 Heldermann Verlag

On the Group of Isometries on a Locally Compact Metric Space

Antonios Manoussos and Polychronis Strantzalos^{∗}

Communicated by W. A. F. Ruppert

Abstract. In the present paper we study conditions under which the group of isometries on a locally compact metric space is locally compact, or acts properly. Keywords: Ellis semigroup, isometry, pointwise convergence, proper action. Mathematics Subject Classification: Primary: 54H20, 37B; Secondary: 54E15, 54H15.

1. Introduction

It is long known from the work of van Dantzig and van der Waerden ([1], cf. also [2, Ch.I, Th.4.7]) that if (X, d) is a connected locally compact metric space then its group of isometries I(X, d), when endowed with the topology of pointwise convergence, is always locally compact and acts properly on X. More recently it was shown by one of the authors ([6]) that the pointwise closure of I(X, d) is locally compact if the space Σ(X) of the connected components of X is quasicompact (compact but not necessarily Hausdorff) with respect to the quotient topology. The question whether I(X, d) is closed in C(X, X) (the space of all continuous selfmaps of X endowed with the topology of pointwise convergence) remained open. In this note we fill this gap (cf. also [4]), i.e., we show that if Σ(X) is quasicompact then I(X, d) coincides with its Ellis’ semigroup, completing the proof of the following:

Theorem. Let (X, d) be a locally compact metric space. Denote by I(X, d) its group of isometries, with the topology of pointwise convergence, and by Σ(X) the space of the connected components of X, endowed with the quotient topology. Then

1. If Σ(X) is not quasicompact, then I(X, d) need not be locally compact, nor act properly on X.

2. If Σ(X) is quasicompact then

ISSN 0949–5932 / $2.50 ◯^{C} Heldermann Verlag (a) I(X, d) is locally compact,

∗ The authors want to thank the referee for his useful and constructive comments.

(b) the action (I(X, d), X) is not always proper, and

(c) the action (I(X, d), X) is proper if X is connected.

For the sake of completeness, we give short and slightly improved proofs of some of the previously published partial results of the authors, these are crucial for a unified proof of the above theorem. Our treatment is based on the sets (x, V_{x}) = {g ∈ I(X, d) : g(x) ∈ V_{x}}, where V_{x} is a neighborhood of x ∈ X. These sets form a neighborhood subbasis at the identity with respect to the topology of pointwise convergence, the natural topology of I(X, d).

2. Generalities

2.1. The following simple examples establish 1 and 2(b) of the above theorem.

Example. Let X = ℤ with the discrete metric. Obviously Σ(X) is not quasicompact. It can be easily seen that I(X, d) is the group of all bijections of ℤ, which is not locally compact with respect to the topology of pointwise convergence, therefore it cannot act properly on a locally compact space.

Example. Let X = Y ∪ {(1, 0)} ⊂ ℝ^{2} where Y = {(0, y) : y ∈ ℝ}, and d = min{1, δ}, where δ denotes the Euclidean metric. As we shall see in §3, by Theorem 3.7, I(X, d) is locally compact; however the action (I(X, d), X) is not proper, because the isotropy group of (1, 0) is not compact, since it contains the translations of Y . So, the action of I(X, d) on X is not proper, even if X has two components.

Since the sets (x, V_{x}) as above form a neighborhood subbasis at the identity in I(X, d), the following condition is necessary for the local compactness of I(X, d):

(a) There exist x_{i} ∈ X, i = 1, …, m such that ⋂_{i=1}^{m}(x_{i}, V_{x_{i}}) is relatively compact in C(X, X).

This condition becomes also sufficient if, in addition, the following condition is satisfied:

(b) I(X, d) is closed in C(X, X).

So, to prove that I(X, d) is locally compact, we have to ensure that both of the above conditions are satisfied.

3. The local compactness of I(X, d)

The following is crucial for the investigation of the conditions 2.1(a) and (b):

3.1. Lemma. Let (X, d) be a locally compact metric space, F ⊆ I(X, d), and

K(F) = {x ∈ X : F (x) = {f(x) : f ∈ F } is relatively compact}.

Then K(F) is an open and closed subset of X.

Proof. Since F is an equicontinuous family of selfmaps of X we see that K(F) is open. It remains to prove that K(F) is closed.

We write S(x, η) = {y ∈ X | d(x, y) < η} for any x ∈ X and η > 0, and S(M, η) = ⋃{S(x, η) | x ∈ M} for subsets M ⊆ X. Let x be a cluster point of K(F) and let η be a positive real such that S(x, 5η) is relatively compact. Choose a point k ∈ K(F) ∩ S(x, η). Then \overline{F (k)} ⊆ S(F (S(x, η)), η) = F (S(x, 2η)), and by the compactness of \overline{F (k)} we can find a finite subset L ⊆ F such that \overline{F (k)} ⊆ L(S(x, 2η)). We show that F (x) is contained in the relatively compact set L(S(x, 5η)). To see this, pick f ∈ F and let g ∈ L such that f(k) ∈ g(S(x, 2η)). Then

\begin{aligned}d(f(x), g(k)) & ≤ d(f(x), f(k)) + d(f(k), g(x)) + d(g(x), g(k)) \\ & = d(x, k) + d(f(k), g(x)) + d(x, k) ≤ 4η \end{aligned}

and therefore

f(x) ∈ S(g(k), 4η) = g(S(k, 4η)) ⊆ g(S(x, 5η)) ⊆ L(S(x, 5η)).

Thus x ∈ K(F) and the proof is finished.

3.2. Remark. In the sequel we assume that Σ(X) is quasicompact in the quotient topology via the natural map q : X → Σ(X). Note that Σ(X) is a T_{1}–space, and need not be Hausdorff. Nevertheless

X is separable, hence second countable; so sequences are adequate in C(X,X). The proof is similar to the lengthy one in [5] (see also [2, Appendix 2]).

3.3. Lemma. Let (X, d) be a locally compact metric space with a quasicompact space of connected components Σ(X). Then condition 2.1(a) is satisfied.

Proof. Let V_{x} be a relatively compact neighborhood of x ∈ X. Then

(x, V_{x}) = {g ∈ I(X, d) : g(x) ∈ V_{x}}

is a neighborhood of the identity in I(X, d). Since x ∈ K((x, V_{x})), K((x, V_{x})) is not empty, and by Lemma 3.1 is open and contains entire components of X. Therefore q(K((x, V_{x}))) is an open subset of Σ(X). Since Σ(X) is quasicompact, there are x_{i}, i = 1, …, m, such that the corresponding q(K((x_{i}, V_{x_{i}})))’ s cover Σ(X). This means that X = ⋃_{i=1}^{m} K((x_{i}, V_{x_{i}})), i.e., the neighborhood F = ⋂_{i=1}^{m}(x_{i}, V_{x_{i}}) of the identity has the property: for every x ∈ X the set F (x) is relatively compact in X. Therefore, by Ascoli’ s theorem, F is relatively compact in C(X, X).

3.4. Now we prove that if Σ(X) is quasicompact then I(X, d) is a closed subspace of C(X, X). Because of Remark 3.2, the elements f of the boundary of I(X, d) in C(X, X) are limits of sequences {f_{n} ∈ I(X, d), n ∈ ℕ}. Obviously, such an f preserves d; so the question is whether f is surjective. If Σ(X) is not quasicompact then this is not always true:

Example. Let X = ℤ with the discrete metric. If f_{n}(z) = z for −n < z < 0, f_{n}(−n) = 0, and f_{n}(z) = z +1 otherwise, then f_{n} → f, where f(z) = z for z < 0, and f(z) = z + 1 for z ≥ 0. Hence each f_{n} is an isometry, but f is not surjective since 0 ∉ f(ℤ).

3.5. Lemma. If Σ(X) is quasicompact and {(f_{n}) : f_{n} ∈ I(X, d)} is a sequence such that f_{n} → f for some selfmap f of X with respect to the topology of pointwise convergence, then f(X) is open and closed in X.

Proof. By Lemma 3.1, it suffices to show that f(X) = K(F), where F = {f_{n}^{−1}, n ∈ ℕ}. Indeed, since d(f_{n}(x), f(x)) = d(x, f_{n}^{−1}(f(x)), we have f_{n}^{−1}(f(x)) → x, so (since X is locally compact) f(x) ∈ K(F), for every x ∈ X. Now, if y ∈ K(F), we may assume f_{n_{k}}^{−1}(y) → x for some x ∈ X, because F (y) is relatively compact in X, hence f(x) = y.

3.6. Proposition. If (X, d) is a locally compact metric space, and Σ(X) is quasicompact, then I(X, d) is closed in C(X, X).

Proof. Let {(f_{n}) : f_{n} ∈ I(X, d)} be a sequence such that f_{n} → f for some selfmap f of X with respect to the topology of pointwise convergence. We prove that f is surjective. Let y ∈ X. We denote by S_{x} the connected component containing x ∈ X, and by S_{n} the component of f_{n}^{−1}(y). If {S_{n}, n ∈ ℕ} has a constant subnet {S_{n_{i}}, i ∈ I}, then S_{n_{i}} = S_{0}, for some S_{0} ∈ Σ(X). Hence S_{f_{n_{i}}^{−1}(y)} = S_{0}, so f_{n_{i}}(S_{0}) = S_{y}, for every i ∈ I. Pick an x ∈ S_{0}, then f_{n_{i}}(x) ∈ S_{y}. But f_{n_{i}}(x) → f(x), so f(x) ∈ S_{y}. By Lemma 3.5 S_{y} ⊆ f(X), hence y ∈ f(X).

Suppose that {S_{n}, n ∈ ℕ} has no constant subnet. By the quasicompactness of Σ(X), there exists a subnet {S_{n_{i}}, i ∈ I} of {S_{n}, n ∈ ℕ} such that S_{n_{i}} → S, for some S ∈ Σ(X). With the above notation, the following is true:

Claim. There exists a subsequence {S_{k}, k ∈ ℕ} of {S_{n}, n ∈ ℕ} such that there are x_{k} ∈ S_{k} with x_{k} → x_{0}, for some x_{0} ∈ X.

Proof. If not, R = (⋃_{n=1}^{∞} S_{n}) \ S is closed in X. Indeed, let {(y_{m}) : y_{m} ∈ R} be a sequence such that y_{m} → y ∈ X. If y_{m} ∈ (⋃_{n=1}^{n_{0}} S_{n}) \ S for m > m_{0}, then a subsequence of {y_{m}, m ∈ ℕ} is contained in some S_{i} for some i ∈ {1, …, n_{0}}, therefore y ∈ S_{i} ⊆ R, as required. If this is not the case, we construct a subsequence {y_{m_{p}}, p ∈ ℕ} of {y_{m}, m ∈ ℕ} in the following way: For S_{1} we choose a point y_{m_{1}} ∈ S_{n_{1}} with n_{1} > 1 and d(y_{m_{1}}, y) < 1, for (⋃_{n=1}^{n_{1}} S_{n}) \ S a point y_{m_{2}} ∈ S_{n_{2}} with n_{2} > n_{1} and d(y_{m_{2}}, y) < \frac{1}{2} , and so on. Obviously, y_{m_{p}} ∈ S_{n_{p}} and y_{m_{p}} → y, a contradiction.

Since S does not meet R, then S ⊆ X \ R. On the other hand X \ R is open (since R is closed in X) and contains entire components (recall that R is a union of components), so S_{n_{i}} ⊆ X \ R, eventually. Therefore S_{n_{i}} = S, a contradiction, since we have assumed that {S_{n}, n ∈ ℕ} has no constant subnet.

According to the Claim, there exists a sequence {(x_{k}) : x_{k} ∈ S_{k}} such that x_{k} → x_{0} ∈ X, where S_{k} = S_{f_{k}^{−1}(y)} = f_{k}^{−1}(S_{y}), from which follows x_{k} = f_{k}^{−1}(y_{k}) for some y_{k} ∈ S_{y}. Then

\begin{aligned}d(y_{k}, f(x_{0})) & ≤ d(y_{k}, f_{k}(x_{0})) + d(f_{k}(x_{0}), f(x_{0})) \\ & = d(f_{k}^{−1}(y_{k}), x_{0}) + d(f_{k}(x_{0}), f(x_{0})) → 0, \end{aligned}

therefore f(x_{0}) ∈ S_{y}, which means that S_{y} ∩ f(X) ≠ Ø and, by Lemma 3.5, S_{y} ⊆ f(X), hence y ∈ f(X), and f is surjective.

3.7. Theorem. If Σ(X) is quasicompact, then I(X, d) is locally compact.

Proof. This assertion follows from Lemma 3.3 and Proposition 3.6, since both conditions 2.1(a) and (b) are satisfied.

4. The properness of the action (I(X, d), X)

In this short section, applying the methods used previously, we give a complete proof of the following:

Proposition. If (X, d) is locally compact and connected, then I(X, d) is locally compact and the action (I(X, d), X) is proper.

Proof. Since X is connected G = I(X, d) is locally compact by Theorem 3.7. So, we have to show that, for every x, y ∈ X, there are neighborhoods U_{x}, U_{y} of x and y respectively such that

(U_{x}, U_{y}) := {g ∈ G : (gU_{x}) ∩ U_{y} ≠ Ø}

is relatively compact in G. Let U_{x} = S(x, ε) and U_{y} = S(y, ε) be such that S(y, 2ε) is relatively compact. Then, for g ∈ (U_{x}, U_{y}) and z ∈ U_{x} with g(z) ∈ U_{y}, we have

d(g(x), (y)) ≤ d(g(x), g(z)) + d(g(z), y) = d(x, z) + d(g(z), y) < 2ε,

therefore g ∈ F = {g ∈ G : g(x) ∈ S(y, 2ε)}. Then x ∈ K(F), and, according to Lemma 3.1, K(F) coincides with the connected space X. From this and Ascoli’s theorem it follows that F is relatively compact in C(X, X). So (U_{x}, U_{y}) ⊆ F is relatively compact in C(X, X), hence in G, because G is closed (cf. Proposition 3.6).

This proves the Proposition and completes the proof of the Theorem in the Introduction.

5. Final Remark

Using the same arguments we can prove that if X is a locally compact metrizable space, then I(X, d) is locally compact for all admissible metrics d, provided that the space Q(X) of the quasicomponents of X is compact with respect to the quotient topology (note that Q(X) is always Hausdorff) (cf. [3]). Recall that the quasicomponent of a point is the intersection of all open and closed sets which contain it. Our exposition is given via Σ(X) because we regard the condition ”Σ(X) is quasicompact” as a topologically more natural condition than ”Q(X) is compact”, although it is more restrictive: There are locally compact metric spaces with compact Q(X) and non quasicompact Σ(X) as the following example shows:

Example. The space of all connected components of the locally compact space

X = ( ⋃_{n=1}^{∞}\{(\frac{1}{n}, y) : y ∈ [−1, 1]\}) ∪ \{(0, y) : y ∈ [−1, 0)\} ∪ ( ⋃_{k=1}^{∞} I_{k}) ⊆ ℝ^{2},

where

I_{k} = \{(0, y) : y ∈ (\frac{1}{k + 1}, \frac{1}{k})\}, k ∈ ℕ^{∗},

is not quasicompact, because the sequence {I_{k}} ⊆ Σ(X) does not have a convergent subsequence in Σ(X). On the contrary, Q(X) is compact, because the quasicomponent of the point (0, −1) consists of the set {(0, y) : y ∈ [−1, 0)} and the intervals I_{k} , k ∈ ℕ^{∗}.

So the quasicompactness of Σ(X) is not necessary for the local compactness of I(X, d).

References

[1] van Dantzig, D., and B. L. van der Waerden, Über metrisch homogene Räume, Abh. Math. Sem. Hamburg 6 (1928), 367–376.

[2] Kobayashi, S., and K. Nomizu, “Foundations of Differential Geometry, Vol. I,” Interscience, New York, 1963.

[3] Manoussos, A., Contribution to the study of D–stable actions, Ph.D. Thesis, University of Athens, Greece (1993) (in Greek).

[4] Manoussos, A., The Ellis semigroup of the full group of isometries of a locally compact metric space, Preprint.

[5] Sierpinsky, W., Sur les espaces métriques localement séparables, Fund. Math. 21 (1933), 107–113.

[6] Strantzalos, P., Actions by Isometries, Lecture Notes in Math. 1375 (1989), 319– 325.

Antonios Manoussos 123 Sapfous Street 176 75 Kallithea Athens Greece amanou@cc.uoa.gr Polychronis Strantzalos Department of Mathematics University of Athens Athens, GR – 157 84 Greece pstrantz@math.uoa.gr

Received August 25, 2000 and in final form March 9, 2002