A CHARACTERISATION OF THE CIRCLE GROUP

WOJCIECH CHOJNACKI

Abstract. We show that if G is a compact connected Abelian group such that, for some n ∈ ℕ and some closed subgroup H of G_{(n)} = {a ∈ G | na = 0}, the set G \ H is disconnected, then G is topologically isomorphic with the circle group 𝕋.

1. Introduction

Let G be a locally compact Abelian group with dual Ĝ. Given n ∈ ℕ, denote by G^{(n)} and G_{(n)} the image and kernel of the homomorphism G ∋ a ↦ na ∈ G, respectively. Given a subset X ⊂ G, let

−X = {a ∈ G | −a ∈ X}.

In agreement with the terminology introduced in [1], G is said to be decomposable if there exists an open subset U ⊂ G such that U ∪ (−U) = G \ G_{(2)} and U ∩ (−U) = ∅.

Let 𝕋 be the circle group, this being the multiplicative group of complex numbers with unit modulus, endowed with the usual topology.

In [1] (see also [2]) the following theorem is established:

Theorem 1. Any decomposable compact connected Abelian group different from a singleton is topologically isomorphic with 𝕋.

This result can be viewed as a characterisation of the circle group. The aim of this paper is to prove the following generalisation of Theorem 1:

Theorem 2. If G is a compact connected Abelian group such that, for some n ∈ ℕ and some closed subgroup H of G_{(n)}, the set G \ H is disconnected, then G is topologically isomorphic with 𝕋.

Of course, the latter theorem can be regarded as yet another characterisation of the circle group.

2. An auxiliary result

This section is devoted to establishing an auxiliary result. We start by fixing notation and recalling some concepts from algebra and topology.

For a set A, denote by #A the cardinality of A, and by id_{A} the identity mapping of A onto itself.

For each n ∈ ℕ, let ℤ(n) be the cyclic group with n elements. Let ℚ be the additive group of rational numbers, equipped with the discrete topology.

If {G_{i}}_{i∈I} is an indexed collection of Abelian groups, we write ∏_{i∈I} G_{i} for the direct product of the G_{i}. If I = {1, …, n}, we also write G_{1} × ⋯ × G_{n} in place of ∏_{i∈I} G_{i}. If 𝔪 is a cardinal number and if, for some fixed G, G_{i} = G for each i ∈ I, where I is a set of cardinality equal to 𝔪, we write G^{𝔪} for ∏_{i∈I} G_{i}.

Let (·, ·) represent the pairing between elements of a locally compact Abelian group and elements of its dual.

1991 Mathematics Subject Classification. Primary 22C05; Secondary 22B05.

Key words and phrases. Compact, connected, Abelian group, circle group.

For a subgroup H of a locally compact Abelian group G, denote by H^{⊥} the annihilator of H in Ĝ, that is, the closed subgroup of Ĝ defined as

H^{⊥} = {χ ∈ Ĝ | (a, χ) = 1 for all a ∈ H}.

For a homomorphism f, designate by ker f the kernel of f.

Given locally compact Abelian groups G and H, and a continuous homomorphism f: G → H, denote by f^{∗}: Ĥ → Ĝ the dual homomorphism defined by

(a, f^{∗}(χ)) = (f(a), χ) (a ∈ G, χ ∈ Ĥ).

An ℕ-indexed projective (or inverse) system of groups is a family {Σ_{p}, π_{p}^{q}}, where, for each p ∈ ℕ, Σ_{p} is a group, and, for all p, q ∈ ℕ with p ≤ q, π_{p}^{q}: Σ_{q} → Σ_{p} is a homomorphism such that the following conditions hold:

(i) π_{p}^{p} = id_{Σ_{p}} for each p ∈ ℕ;

(ii) π_{p}^{q}π_{q}^{r} = π_{p}^{r} for all p, q, r ∈ ℕ with p ≤ q ≤ r.

If in addition each Σ_{p} is a topological group and each π_{p}^{q} is a continuous homomorphism, then {Σ_{p}, π_{p}^{q}} is called a topological projective system of groups. The projective limit of such a system \{Σ_{p}, π_{p}^{q}\} is the group \underleftarrow{lim}\{Σ_{p}, π_{p}^{q}\} defined as

\underleftarrow{lim}\{Σ_{p}, π_{p}^{q}\} = \{\{a_{p}\}_{p∈ℕ} ∈ ∏_{p∈ℕ} Σ_{p} | π_{p}^{q}(a_{q}) = a_{p} for all p, q ∈ ℕ with p ≤ q\}.

For each p ∈ ℕ, let π_{p}: \underleftarrow{lim}\{Σ_{p}, π_{p}^{q}\} → Σ_{p} be the homomorphism defined as the restriction to \underleftarrow{lim}\{Σ_{p}, π_{p}^{q}\} of the canonical projection of ∏_{p∈ℕ} Σ_{p} onto Σ_{p}. The π_{p} are compatible with the π_{p}^{q} in the sense that π_{p}^{q}π_{q} = π_{p} for all p, q ∈ ℕ with p ≤ q. If \{Σ_{p}, π_{p}^{q}\} is a topological projective system of groups, then \underleftarrow{lim}\{Σ_{p}, π_{p}^{q}\} can be given the weakest topology making all the projection maps π_{p} continuous. This topology is just the relativised topology from the direct product. Any family of sets of the form π_{p}^{−1}(U_{p}), where p ranges over an arbitrarily fixed infinite subset of ℕ and U_{p} is an open subset of Σ_{p}, is a base for the topology of \underleftarrow{lim}\{Σ_{p}, π_{p}^{q}\}.

It is easy to check that projective limits satisfy the universal property that if Γ is another (topological) group with a family of (continuous) homomorphisms σ_{p}: Γ → Σ_{p} satisfying π_{p}^{q}σ_{q} = σ_{p} for all p, q ∈ ℕ with p ≤ q, then there is a unique (continuous) homomorphism σ: Γ → \underleftarrow{lim}\{Σ_{p}, π_{p}^{q}\} satisfying π_{p}σ = σ_{p} for all p ∈ ℕ.

Proposition. Let {Σ_{p}, π_{p}^{q}} be an ℕ-indexed topological projective system of groups such that:

(i) there exists l ∈ ℕ such that, for each p ∈ ℕ, Σ_{p} is topologically isomorphic with 𝕋^{l} by means of a homomorphism τ_{p}: Σ_{p} → 𝕋^{l};

(ii) for each p, q ∈ ℕ with p ≤ q, π_{p}^{q} has the form

π_{p}^{q} = τ_{p}^{−1}τ_{p}^{q}τ_{q},

where τ_{p}^{q}: 𝕋^{l} → 𝕋^{l} is the homomorphism

τ_{p}^{q}: (t_{1}, …, t_{l}) ↦ (t_{1}^{n_{p}^{q}(1)}, …, t_{l}^{n_{p}^{q}(l)})  (1)

for some n_{p}^{q}(1), …, n_{p}^{q}(l) ∈ ℤ \ {0}; (iii) if l = 1, then lim_{q→∞} |n_{p}^{q}(1)| = +∞ for each p ∈ ℕ.

Then Σ = \underleftarrow{lim}\{Σ_{p}, π_{p}^{m}\} has the property that, for each n ∈ ℕ and each closed subgroup Γ of Σ_{(n)}, Σ \ Γ is connected.

Proof. Without loss of generality, we may assume that Σ_{p} = 𝕋^{l} and τ_{p} = id_{Σ_{p}} for all p ∈ ℕ, and π_{p}^{q} = τ_{p}^{q} for all p, q ∈ ℕ with p ≤ q. That {Σ_{p}, π_{p}^{q}} is an inverse system of groups now means that, for each j = 1, …, l, n_{p}^{p}(j) = 1 for each p ∈ ℕ, and n_{p}^{q}(j)n_{q}^{r}(j) = n_{p}^{r}(j) for all p, q, r ∈ ℕ with p ≤ q ≤ r. Suppose, on the contrary, that, for some n ∈ ℕ and some closed subgroup Γ of Σ_{(n)}, Σ \ Γ can be decomposed into two non-empty disjoint open (in Σ \ Γ) sets U_{1} and U_{2}. Since Γ is a closed subgroup of Σ_{(n)} and since Σ_{(n)} is a closed subgroup of Σ, U_{1} and U_{2} are open subsets of Σ. For each i = 1, 2, choose x_{i} ∈ U_{i} arbitrarily, and next select an open neighbourhood of x_{i} of the form π_{p_{i}}^{−1}(V_{i}), contained in U_{i}, with p_{i} ∈ ℕ and V_{i} an open subset of Σ_{p_{i}}. We claim that there exists q ∈ ℕ with q ≥ p_{i} for each i = 1, 2 such that, for some a_{1}, a_{2} ∈ Σ_{q},

π_{q}^{−1}({a_{i}}) ⊂ U_{i} for each i = 1, 2  (2)

and such that a_{1} and a_{2} can be joined by a closed arc I wholly contained in Σ_{q} \ π_{q}(Γ).

First consider the case l > 1. Since, for each p ∈ ℕ, π_{p}(Γ) ⊂ π_{p}(Σ_{(n)}) ⊂ (Σ_{p})_{(n)} and since, clearly, (Σ_{p})_{(n)} is isomorphic with ℤ(n)^{l}, we have

#π_{p}(Γ) ≤ n^{l}.  (3)

Select q ∈ ℕ so that q ≥ p_{i} for each i = 1, 2. In view of (3), π_{q}(Γ) is finite. Hence, since Σ_{q} is an l-dimensional torus with l > 1, Σ_{q} \ π_{q}(Γ) is arc-wise connected. For each i = 1, 2, set a_{i} = π_{q}(x_{i}). Clearly, a_{1} and a_{2} can be linked by a closed arc I wholly contained in Σ_{q} \ π_{q}(Γ). Furthermore

π_{p_{i}}^{q}(a_{i}) = π_{p_{i}}^{q}(π_{q}(x_{i})) = π_{p_{i}}(x_{i}),

whence π_{q}^{−1}({a_{i}}) ⊂ π_{p_{i}}^{−1}({π_{p_{i}}(x_{i})}). As π_{p_{i}}(x_{i}) ∈ V_{i} and π_{p_{i}}^{−1}(V_{i}) ⊂ U_{i}, (2) is implied.

We now pass to the case l = 1. Using condition (iii) of the statement, choose q ∈ ℕ so that |n_{p_{i}}^{q}(1)| > n for each i = 1, 2. Being a subgroup of the cyclic group (Σ_{p})_{(n)}, π_{q}(Γ) is cyclic. Thus Σ_{q} \ π_{q}(Γ) consists of open arcs, each of which has length equal to 2π/#π_{q}(Γ). Let J be one of these arcs. In view of (3), the length of J is no smaller than 2π/n. Now, for each i = 1, 2, ker π_{p_{i}}^{q} is the cyclic group of all roots of unity of order |n_{p_{i}}^{q}(1)|. Therefore, for each i = 1, 2, every coset of ker π_{p_{i}}^{q} consists of points evenly distributed around the circle, with the angular distance between any pair of two closest points being equal to 2π/|n_{p_{i}}^{q}(1)|. As 2π/|n_{p_{i}}^{q}(1)| < 2π/n, we see that, for each i = 1, 2, every coset of ker π_{p_{i}}^{q} has a point in common with J. For each i = 1, 2, pick a_{i} ∈ (ker π_{p_{i}}^{q})π_{q}(x_{i}) ∩ J, where (ker π_{p_{i}}^{q})π_{q}(x_{i}) denotes the coset of ker π_{p_{i}}^{q} containing π_{q}(x_{i}). It is readily seen that π_{p_{i}}^{q}(a_{i}) = π_{p_{i}}(x_{i}), and hence, as before, we obtain (2). Taking for I a subarc of J having the a_{i} for endpoints finally establishes the claim.

The inclusion I ⊂ Σ_{q} \ π_{q}(Γ) now implies that π_{q}^{−1}(I) ⊂ Σ \ Γ. In view of the form of the bases for the topology of a projective limit, π_{q}^{−1}(I) can be covered by open sets π_{r}^{−1}(W_{r}), where r ∈ ℕ satisfies r ≥ q and W_{r} is an open subset of Σ_{r} such that either π_{r}^{−1}(W_{r}) ⊂ U_{1} or π_{r}^{−1}(W_{r}) ⊂ U_{2} holds. Since π_{q}^{−1}(I) is compact, we can choose a finite subcover π_{r_{1}}^{−1}(W_{r_{1}}), …, π_{r_{d}}^{−1}(W_{r_{d}}). Fix s ∈ ℕ so that s ≥ r_{j} for all j = 1, …, d. For each j = 1, …, d, set W′_{j} = (π_{r_{j}}^{s})^{−1}(W_{r_{j}}). Noting that π_{r_{j}}^{−1}(W_{r_{j}}) = π_{s}^{−1}(W′_{j}), let, for each i = 1, 2, Z_{i} be the union of all those W′_{j} for which π_{s}^{−1}(W′_{j}) ⊂ U_{i}. Clearly, Z_{1} and Z_{2} are open disjoint subsets of Σ_{s} such that (π_{q}^{s})^{−1}(I) ⊂ Z_{1} ∪ Z_{2}. Since π_{q}^{s} is a covering map, there is a continuous map f: I → Σ_{s} such that π_{q}^{s} ∘f = id_{Σ_{q}}. Now, clearly, f(I) ⊂ (π_{q}^{s})^{−1}(I), and, since f(I) is connected, we have either f(I) ⊂ Z_{1} or f(I) ⊂ Z_{2}, and consequently either π_{s}^{−1}(f(I)) ⊂ U_{1} or π_{s}^{−1}(f(I)) ⊂ U_{2}. But π_{s}^{−1}(f({a_{i}})) ⊂ π_{q}^{−1}({a_{i}}) and therefore either

π_{q}^{−1}({a_{i}}) ∩ U_{1} ≠ ∅ for each i = 1, 2

or

π_{q}^{−1}({a_{i}}) ∩ U_{2} ≠ ∅ for each i = 1, 2.

This, however, is incompatible with (2), as U_{1} and U_{2} are disjoint. The contradiction obtained establishes the result. □

3. Proof of the main result

This section is devoted to establishing Theorem 2.

Proof of Theorem 2. Let G be a compact connected Abelian group for which there exist n ∈ ℕ and a closed subgroup H ⊂ G_{(n)} such that G\H is disconnected. Then, necessarily, both G and Ĝ are different from a singleton. By the connectedness of G, Ĝ is torsion free (cf. [3, §24.25]). Let {χ_{α}}_{α∈A} be a maximal collection of independent elements of Ĝ. As is known, #A does not depend on the particular choice of a maximal family of independent members of Ĝ, and defines the (torsion-free) rank of Ĝ. By the maximality of {χ_{α}}_{α∈A}, for each χ ∈ Ĝ there exist n(χ) ∈ ℤ and an A-indexed family of integers {n_{α}(χ)}_{α∈A} such that: (i) n_{α}(χ) ≠ 0 for only finitely many α ∈ A; (ii) the equality n(χ)χ = ∑_{α∈A} n_{α}(χ)χ_{α} holds. By the independency of the χ_{α}, n(χ) can be taken to be non-zero so that—in particular—for each α ∈ A the rational number n_{α}(χ)/n(χ) makes sense; moreover, this number depends only on χ. One verifies at once that, for each α ∈ A, the function ρ_{α}: χ ↦ n_{α}(χ)/n(χ) is a homomorphism from Ĝ into ℚ. Since Ĝ is torsion free, we have

⋂_{α∈A} ker ρ_{α} = ∅.  (4)

Let P_{fin}(A) be the set of all finite subsets of A. For each A = {α_{1}, …, α_{k}} in P_{fin}(A), define a homomorphism ρ_{A} : Ĝ → ℚ^{k} by setting

ρ_{A} = (ρ_{α_{1}}, …, ρ_{α_{k}}).

Observe that, for each A ∈ P_{fin}(A), the dual of ρ_{A}(Ĝ) is topologically isomorphic with (ker ρ_{A})^{⊥}, where the annihilator is taken in the dual of Ĝ identified with G. In view of (4),

G = ⋃_{A∈P_{fin}(A)}(ker ρ_{A})^{⊥},

and hence

G \ H = ⋃_{A∈P_{fin}(A)}(ker ρ_{A})^{⊥} \ (H ∩ (ker ρ_{A})^{⊥}).

Taking into account that G \ H is disconnected, we immediately deduce from the last equality that for each A ∈ P_{fin}(A) there exists B ∈ P_{fin}(A) with A ⊂ B such that (ker ρ_{B})^{⊥} \ (H ∩ (ker ρ_{B})^{⊥}) is disconnected.

Fix A ∈ P_{fin}(A) arbitrarily and choose B = {α_{1}, …, α_{l}} in P_{fin}(A) so that A ⊂ B and (ker ρ_{B})^{⊥} \ (H ∩ (ker ρ_{B})^{⊥}) is disconnected. For each p ∈ ℕ, let K_{p} be the cyclic subgroup of ℚ given by

K_{p} = {m/p! | m ∈ ℤ}

and let L_{p} = ρ_{B}(Ĝ) ∩ (K_{p})^{l}. It is clear that L_{p} ⊂ L_{p+1} for each p ∈ ℕ and that

ρ_{B}(Ĝ) = ⋃_{p=1}^{∞} L_{p}.  (5)

Since, for each p ∈ ℕ, L_{p} is a subgroup of the direct product of l copies of the cyclic group K_{p}, it follows that L_{p} is a direct product of cyclic groups, and hence is up to isomorphism determined by its rank. It is apparent that

L_{p} ⊂ (ρ_{α_{1}}(Ĝ) ∩ K_{p}) × ⋯ × (ρ_{α_{l}}(Ĝ) ∩ K_{p}).  (6)

From

\begin{aligned}ρ_{B}(χ_{α_{1}}) & = (1, 0, …, 0), \\ ρ_{B}(χ_{α_{2}}) & = (0, 1, …, 0), \end{aligned}

…
ρ_{B}(χ_{α_{l}}) = (0, 0, …, 1),

we infer that the rank of L_{p} is no smaller than l. On the other hand, for each i = 1, …, l, the group ρ_{α_{i}}(Ĝ) ∩ K_{p} is cyclic, and so the rank of (ρ_{α_{1}}(Ĝ) ∩ K_{p}) × ⋯ × (ρ_{α_{l}}(Ĝ) ∩ K_{p}) is equal to l. Coupling this with (6), we see that

L_{p} = (ρ_{α_{1}}(Ĝ) ∩ K_{p}) × ⋯ × (ρ_{α_{l}}(Ĝ) ∩ K_{p}).

For any p, q ∈ ℕ with p ≤ q, let i_{p}^{q} be the canonical embedding of L_{p} into L_{q}. Clearly, i_{p}^{p} = id_{L_{p}} and hence i_{p}^{p}^{∗} = idL̂_{p} for each p ∈ ℕ, and also i_{q}^{r}i_{p}^{q} = i_{p}^{r} and hence i_{p}^{q}^{∗}i_{q}^{r}^{∗} = i_{p}^{r}^{∗}

for all p, q, r ∈ ℕ with p ≤ q ≤ r. Thus {L̂_{p}, i_{p}^{q}^{∗}} is a topological projective system of groups.

For each p ∈ ℕ, let i_{p} be the canonical embedding of L_{p} into ρ_{B}(Ĝ). Clearly, i_{p} = i_{q}i_{p}^{q} and so i_{p}^{∗} = i_{p}^{q}^{∗}i_{q}^{∗} for any p, q ∈ ℕ with p ≤ q. By the universal property of projective limits, there is a unique continuous homomorphism σ: (ker ρ_{B})^{⊥} → \underleftarrow{lim}\{L̂_{p}, i_{p}^{q}^{∗}\} satisfying π_{p}σ = i_{p}^{∗} for all p ∈ ℕ; here, of course, π_{p} stands for the projection map from \underleftarrow{lim}\{L̂_{p}, i_{p}^{q}^{∗}\} onto L̂_{p}. From (5) we deduce that σ is injective. Since the i_{p} are injective, it follows that the σ_{p} are surjective, and consequently that σ is surjective. Thus σ is a continuous bijection, and as (ker ρ_{B})^{⊥} is compact, σ is a homeomorphism and hence a topological isomorphism from \underleftarrow{lim}\{L̂_{p}, i_{p}^{q}^{∗}\} onto (ker ρ_{B})^{⊥}.

For each p ∈ ℕ and each j = 1, …, l, let χ_{j,p} ∈ Ĝ be a generator of K_{p} ∩ ρ_{α_{j}}(Ĝ). For any p, q ∈ ℕ with p ≤ q ∈ ℕ and for each j = 1, …, l, let n_{p}^{q}(j) ∈ ℤ \ {0} be such that

ρ_{α_{j}}(χ_{j,p}) = n_{p}^{q}(j) ρ_{α_{j}}(χ_{j,q}).

For each j arbitrarily fixed, we can arrange all the n_{p}^{p+1}(j) to be positive by replacing, if necessary, χ_{j,p} by −χ_{j,p} successively as p increases. Since

n_{p}^{q}(j) = ∏_{r=p}^{q−1} n_{r}^{r+1}(j),  (7)

all the n_{p}^{q}(j) will then be positive too.

For each p ∈ ℕ, let j_{p}: ℤ^{l} → L_{p} be the homomorphism

j_{p}: (a_{1}, …, a_{l}) ↦ (a_{1}ρ_{α_{1}}(χ_{1,p}), …, a_{l}ρ_{α_{l}}(χ_{l,p})).

Since j_{p} is bijective, the dual homomorphism j_{p}^{∗}: L̂_{p} → 𝕋^{l} is a topological isomorphism. Thus {L̂_{p}, i_{p}^{q}^{∗}} satisfies condition (i) from Proposition.

For any p, q ∈ ℕ with p ≤ q ∈ ℕ, let j_{p}^{q}: ℤ^{l} → ℤ^{l} be the homomorphism

j_{p}^{q}: (a_{1}, …, a_{l}) ↦ (n_{p}^{q}(1)a_{1}, …, n_{p}^{q}(l)a_{l}).

It is easily verified that i_{p}^{q}j_{p} = j_{q}j_{p}^{q} and hence j_{p}^{∗}i_{p}^{q}^{∗} = j_{p}^{q}^{∗}j_{q}^{∗}. Since j_{p}^{q}^{∗} can be identified with τ_{p}^{q} given by (1), we see that {L̂_{p}, i_{p}^{q}^{∗}} satisfies condition (ii) from Proposition.

Assume now that #A ≥ 2. Then #B ≥ 2 and since \underleftarrow{lim}\{L̂_{p}, i_{p}^{q}^{∗}\} is topologically isomorphic with (ker ρ_{B})^{⊥} and

H ∩ (ker ρ_{B})^{⊥} ⊂ ((ker ρ_{B})^{⊥})_{(n)},

it follows from Proposition that (ker ρ_{B})^{⊥} \(H ∩(ker ρ_{B})^{⊥}) is connected, a contradiction. Therefore #A = 1 and consequently, in view of the arbitrariness of A, A is a singleton. In particular, A = B = A, l = 1, and, by (4), (ker ρ_{B})^{⊥} = G.

Repeating the argument, we infer that {L̂_{p}, i_{p}^{q}^{∗}} does not satisfy condition (iii) of Proposition. Now either there exists p_{0} ∈ ℕ such that n_{p}^{p+1}(1) = 1 for all p ∈ ℕ with p ≥ p_{0}, or there is a sequence {p_{k}}_{k∈ℕ} in ℕ diverging to infinity such that n_{p_{k}}^{p_{k}+1}(1) = 1 for all k ∈ ℕ. Using (7), it is easy to see that the first possibility holds precisely when condition (iii) of Proposition is met. This implies that the second possibility holds, and now appealing to (7) again, we find that n_{p}^{q}(1) = 1 for all p, q ∈ ℕ with p, q ≥ p_{0}. Consequently, \underleftarrow{lim}\{Σ_{p}, π_{p}^{q}\} reduces to a group topologically isomorphic with 𝕋. As \underleftarrow{lim}\{Σ_{p}, π_{p}^{q}\} is topologically isomorphic with (ker ρ_{B})^{⊥} and as the latter group coincides with G, we finally conclude that G is topologically isomorphic with 𝕋. □

References

1. W. Chojnacki, Group representations of bounded cosine functions, J. Reine Angew. Math. 478 (1996), 61–84.

2. , A new proof of a theorem concerning decomposable groups, Glasnik Mat. Ser. III 33 (1998), 13–17.

3. E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, vol. 1, Springer-Verlag, Berlin, 1963.

Instytut Matematyki Stosowanej i Mechaniki, Uniwersytet Warszawski, ul. Banacha 2, 02-097 Warszawa, Poland

   E-mail address: wojtekch@appli.mimuw.edu.pl

Department of Computer Science, University of Adelaide, Adelaide, SA 5005, Australia

   E-mail address: wojtek@cs.adelaide.edu.au