New York Journal of Mathematics

New York J. Math. 13 (2007) 159–174.

The Choquet–Deny theorem and distal properties of totally disconnected locally compact groups of

polynomial growth

Wojciech Jaworski and C. Robinson Edward Raja

Abstract. We obtain sufficient and necessary conditions for the Choquet– Deny theorem to hold in the class of compactly generated totally disconnected locally compact groups of polynomial growth, and in a larger class of totally disconnected generalized F C-groups. The following conditions turn out to be equivalent when G is a metrizable compactly generated totally disconnected locally compact group of polynomial growth:

(1) The Choquet–Deny theorem holds for G.

(2) The group of inner automorphisms of G acts distally on G.

(3) Every inner automorphism of G is distal.

(4) The contraction subgroup of every inner automorphism of G is trivial.

(5) G is a SIN group. We also show that for every probability measure μ on a totally disconnected compactly generated locally compact second countable group of polynomial growth, the Poisson boundary is a homogeneous space of G, and that it is a compact homogeneous space when the support of μ generates G.

Contents

1. Introduction 160 2. Distal properties of totally disconnected locally compact groups 161 3. The Choquet–Deny theorem 166 4. On boundaries of random walks 168 References 173

Received January 24, 2007. Revised June 3, 2007. Mathematics Subject Classification. 60B15, 43A05, 60J50, 22D05, 22D45. Key words and phrases. Choquet–Deny theorem, totally disconnected groups, polynomial

growth, distal, random walks, Poisson boundary. The first author was supported by an NSERC Grant.

159^{ISSN 1076-9803/07}

1. Introduction

Let μ be a regular Borel probability measure on a locally compact group G. A bounded Borel function h: G → ℂ is called μ-harmonic if it satisfies

(1.1)  h(g) = ∫_{G} h(gg′) μ(dg′), g ∈ G.

The classical Choquet–Deny theorem asserts that when G is abelian then every bounded continuous μ-harmonic function is constant on the (left) cosets of the smallest closed subgroup, G_{μ}, containing the support of μ.

The Choquet–Deny theorem remains true for many nonabelian locally compact groups, e.g., 2-step nilpotent groups [10], nilpotent [SIN] groups [14], and compact groups. But it does not hold for all groups. If the theorem holds for a probability measure μ then G_{μ} must necessarily be an amenable subgroup [7, 30]. It follows that groups for which the theorem is valid are necessarily amenable. However, the theorem is not true for every amenable group [23].

The stronger condition, that G have polynomial growth, is sufficient for the theorem to hold when G is a finitely generated (discrete) group [23, 21]. When G is finitely generated and solvable then the theorem holds if and only if G has polynomial growth [21]. In general, the theorem fails for discrete groups of polynomial growth that are not finitely generated, in particular, it is not true for locally finite groups [23]. It appears that the largest class of discrete groups known today for which the Choquet–Deny theorem is true is the class of FC-hypercentral groups [15]. This class is a proper subclass of the class of discrete groups of polynomial growth, while finitely generated FC-hypercentral groups are precisely the finitely generated groups of polynomial growth. We do not know of any discrete groups for which the Choquet–Deny theorem is true and which are not FC-hypercentral.

A probability measure μ on a locally compact group G is called spread out if for some n the convolution power μ^{n} is nonsingular. With the restriction that μ be spread out the Choquet–Deny theorem holds for all locally compact nilpotent groups [2, 19] and for compactly generated locally compact groups of polynomial growth [19]. When G is almost connected, then G has polynomial growth if and only if the Choquet–Deny theorem holds for every spread out measure [19]. The same is true when G is a Zariski-connected p-adic algebraic group [27, Theorem 4.2]. While it remains an open question whether the spread out assumption can be disposed of when G is nilpotent,^{1} it is known that the Choquet–Deny theorem is not true for arbitrary probability measures on compactly generated locally compact groups of polynomial growth [16, Remark 3.15].

The main goal of the present article is to obtain necessary and sufficient conditions for the validity of the Choquet–Deny theorem in the class of compactly generated totally disconnected locally compact groups of polynomial growth, and in a larger class of totally disconnected ‘generalized F C-groups’ [3, 24]. It turns out that the key to finding such conditions is a study of distal properties of totally disconnected groups. This motivates our investigations in the next section, which can also be of quite independent interest. The Choquet–Deny theorem for generalized F C-groups is discussed in Section 3. In Section 4 we remark on the structure of boundaries of random walks on compactly generated totally disconnected groups of polynomial growth and certain related groups.

1 The recently published proof [28] is wrong (the implications (i)⇒(ii) and (i)⇒(iii) in the key Lemma 2.5 are false).

2. Distal properties of totally disconnected locally compact groups

Let G be a Hausdorff topological group and Γ a subgroup of Aut(G), the group of topological automorphisms of G. We will say that Γ is distal (or acts distally on G) if for any x ∈ G − {e}, the identity element e is not in the closure of the orbit Γ x = {γ(x) ; γ ∈ Γ }. A single automorphism γ ∈ Aut(G) will be called distal if the subgroup, ⟨γ⟩, it generates acts distally on G. An element g of G will be called distal if the corresponding inner automorphism γ(·) = g · g^{−1} is distal. We will say that G is distal if the group Inn(G) of inner automorphisms of G acts distally on G.

Trivially, if G is distal then every g ∈ G is distal. While the converse is not true in general, Rosenblatt [29] proved that when G is an almost connected locally compact group then G is distal if and only if every g ∈ G is distal; moreover G is distal if and only if it has polynomial growth. According to [26] this remains true also for certain classes of p-adic Lie groups. However, there are many locally compact groups of polynomial growth that are not distal. For example, the semidirect product K ×_{τ} ℤ where K is a nontrivial compact metric group and τ is an ergodic automorphism of K, will never be distal.

Given γ ∈ Aut(G) the contraction subgroup of γ is the subgroup C(γ) = {x ∈ G; lim_{n→∞} γ^{n}(x) = e}. When γ is the inner automorphism γ(·) = g · g^{−1}, we will write C(g) for C(γ). Obviously, if τ ∈ Aut(G) is distal then C(τ) = C(τ^{−1}) = {e}. When G is a Lie group, the three conditions: Γ is distal; every γ ∈ Γ is distal; and, C(γ) = {e} for every γ ∈ Γ, are equivalent for every subgroup Γ of Aut(G) [1].

Recall that a subgroup Γ of Aut(G) is equicontinuous (at e) if and only if G admits a neighbourhood base at e consisting of neighbourhoods that are invariant under Γ. When G is locally compact and totally disconnected then Γ is equicontinuous if and only if compact open subgroups invariant under Γ form a neighbourhood base at e. Equicontinuous automorphism groups are obviously distal. A SIN group is a topological group G for which Inn(G) is equicontinuous. SIN groups are distal but, in general, distal groups are not SIN groups (e.g., a nilpotent group need not be SIN but every nilpotent group is distal [29]).

Our goal in this section is to prove that for a class of compactly generated totally disconnected locally compact groups, including groups of polynomial growth, the four conditions: G is distal; every g ∈ G is distal; C(g) = {e} for every g ∈ G; and, G is SIN, are equivalent. In the following section we will show that for this class of groups the four conditions and the condition that G have polynomial growth, are equivalent to the condition that the Choquet–Deny theorem hold for G.

Some of the recent results of Baumgartner and Willis on contractions subgroups [4], based on Willis’ theory of tidy subgroups [32], play a key role in our argument. These results are proven for metrizable groups, hence, in many of our results we need to assume metrizability.^{2}

Proposition 2.1. If G is a totally disconnected metrizable locally compact group then for every τ ∈ Aut(G) the following conditions are equivalent:

2See note added in proof.

(i) τ is distal.

(ii) C(τ) = C(τ^{−1}) = {e}.

(iii) For every compact open subgroup U there exists k = 0, 1, … such that

τ(⋂_{i=0}^{k} τ^{i}(U)) = ⋂_{i=0}^{k} τ^{i}(U). (iv) ⟨τ⟩ is equicontinuous.

Proof. The only nonobvious implication in the chain (i)⇒(ii)⇒(iii)⇒(iv)⇒(i) is (ii)⇒(iii). Let U be a compact open subgroup. Since C(τ) is closed, by [4, Theorem 3.32] there exists k such that V = ⋂_{i=0}^{k} τ^{i}(U) is tidy for τ. But as C(τ) = C(τ^{−1}) = {e}, [4, Proposition 3.24] implies that s(τ) = s(τ^{−1}) = 1 where s: Aut(G) → ℕ is the scale function. Since s(τ) = [τ(V) : V ∩ τ(V)] and s(τ^{−1}) = [τ^{−1}(V) : V ∩ τ^{−1}(V)], so τ(V) = V. □

Lemma 2.2. Let Γ be a subgroup of Aut(G) where G is a totally disconnected metrizable locally compact group. If τ_{1}, τ_{2}, …, τ_{n} ∈ Aut(G) are distal and for every j = 1, 2, …, n, [τ_{j}, ⟨Γ ∪ {τ_{1}, …, τ_{j−1}}⟩] ⊆ ⟨Γ ∪ {τ_{1}, …, τ_{j−1}}⟩, then for every compact open subgroup U invariant under Γ there exists a compact open subgroup V ⊆ U invariant under ⟨Γ ∪ {τ_{1}, …, τ_{n}}⟩.

Proof. It is clear that the lemma follows by induction once it is verified for n = 1. So we suppose that n = 1.

By Proposition 2.1 there exists k such that V = ⋂_{i=0}^{k} τ_{1}^{i}(U) satisfies τ_{1}(V) = V. It is enough to show that γ(V) = V for every γ ∈ Γ. But our assumption implies that [τ_{1}^{i}, Γ] ⊆ Γ for every i = 0, 1, …. Hence, given γ ∈ Γ we obtain γ(V) = ⋂_{i=0}^{k}(γτ_{1}^{i})(U) = ⋂_{i=0}^{k}(τ_{1}^{i}γ[γ, τ_{1}^{i}])(U) = ⋂_{i=0}^{k} τ_{1}^{i}(U) = V. □

Lemma 2.3. Let Γ be a subgroup of Aut(G) where G is a totally disconnected metrizable locally compact group. Suppose that every γ ∈ Γ is distal and that Γ has a normal equicontinuous subgroup Γ_{1} with the quotient Γ/Γ_{1} containing a polycyclic subgroup of finite index. Then Γ is equicontinuous.

Proof. Let Ω be a neighbourhood of e. Denote by P the polycyclic subgroup of finite index in Γ/Γ_{1} and let P_{0} = P, P_{1} = [P, P], P_{2} = [P_{1}, P_{1}], …, P_{m} = {Γ_{1}} be the derived series for P. Write π for the canonical homomorphism π: Γ → Γ/Γ_{1} and put P̂_{j} = π^{−1}(P_{j}) for j = 0, 1, ..., m.

Suppose that for some j = 1, 2, ..., m, V ⊆ Ω is a compact open subgroup invariant under P̂_{j}. We will show that there is then a compact open subgroup W ⊆ V invariant under P̂_{j−1}. Now, since P is polycyclic, P_{j−1} is generated by a finite set {p_{1}, …, p_{n}}. For every i = 1, 2, …, n find τ_{i} ∈ P̂_{j−1} with p_{i} = π(τ_{i}). Applying Lemma 2.2 to P̂_{j} and τ_{1}, …, τ_{n} we conclude that there is a compact open subgroup W ⊆ V invariant under ⟨P̂_{j} ∪ {τ_{1}, …, τ_{n}}⟩ = P̂_{j−1}.

Our assumption is that there is a compact open subgroup V ⊆ Ω such that γ(V) = V for every γ ∈ P̂_{m} = Γ_{1}. With the aid of the preceding paragraph it then follows that there is a compact open subgroup W ⊆ Ω invariant under P̂_{0}. Next, since P = P_{0} has finite index in Γ/Γ_{1}, P̂_{0} has finite index in Γ. Hence, the intersection U = ⋂_{γ∈Γ} γ(W) is a compact open subgroup invariant under Γ and contained in Ω. □

Corollary 2.4. Let G be a totally disconnected metrizable locally compact group. If a subgroup Γ of Aut(G) contains a polycyclic subgroup of finite index then the following conditions are equivalent:

(i) Γ is distal.

(ii) Every γ ∈ Γ is distal.

(iii) Γ is equicontinuous.

As the following examples show, ‘polycyclic’ in Corollary 2.4 cannot be replaced by ‘solvable’. In fact, the three conditions can be different for countable abelian groups of automorphisms. We do not know if ‘polycyclic’ can be replaced by ‘finitely generated solvable’.

Example 2.5. Let φ: ℝ → 𝕋 denote the function φ(t) = e^{2πit} and let H be any infinite subgroup of φ(ℚ). Note that every h ∈ H has finite order. Let G be the totally disconnected compact abelian group G = ℤ_{2}^{H}. H acts on G by left translations: (hf)(x) = f(h^{−1}x) (h ∈ H, f ∈ G, x ∈ H). Let Γ be the resulting subgroup of Aut(G). Then every element of Γ is distal because it has finite order. However, Γ is not distal. Indeed, let f ∈ G be the function f(x) = δ_{1x} and let U be any neighbourhood of e in G. Then for some finite subset F ⊆ H, U contains the set {g ∈ G; g(x) = 0 for every x ∈ F }. Hence, if h ∈ H − F then (hf)(x) = 0 for every x ∈ F, i.e., hf ∈ U.

Thus for a countable abelian group of automorphisms (ii) does not imply (i) (nor (iii)).

Example 2.6. Let for j ∈ ℤ, G_{j} = {x ∈ ℤ_{2}^{ℤ}; x_{i} = 0 for every i ≤ j} and let G = ⋃_{j∈ℤ} G_{j}. There is a locally compact totally disconnected group topology on G in which the subgroups G_{j}, j ∈ ℤ, form a neighbourhood base at e (and are compact open). Given j ∈ ℕ define τ_{j} ∈ Aut(G) by τ_{j}(x) = y where y_{i} = x_{i} for i ∈ ℤ − {±j}, y_{j} = x_{−j}, and y_{−j} = x_{j}. The subgroup Γ of Aut(G) generated by τ_{j}, j ∈ ℕ, is then abelian and distal: if x ≠ e and j is the smallest integer with x_{j} ≠ 0, then τ(x) ∉ G_{|j|} for any τ ∈ Γ. However, Γ is not equicontinuous because if it were, there would exist k ≥ 0 with τ(x) ∈ G_{0} for every x ∈ G_{k} and τ ∈ Γ. However, if x = (δ_{k+1 n})_{n∈ℤ} then x ∈ G_{k}, but (τ_{k+1}(x))_{−k−1} = x_{k+1} = 1, so that τ_{k+1}(x) ∉ G_{0}.

Thus for a countable abelian group of automorphisms (i) does not imply (iii).

Theorem 2.7. Suppose that a totally disconnected metrizable locally compact group G admits an open normal SIN subgroup N such that the quotient G/N contains a polycyclic subgroup of finite index. Then the following conditions are equivalent:

(i) G is distal.

(ii) Every g ∈ G is distal.

(iii) G is SIN.

Proof. (ii)⇒(iii): Let Γ = Inn(G) and α : G → Γ be the canonical homomorphism. Put Γ_{1} = α(N). Γ_{1} is a normal subgroup of Γ and Γ/Γ_{1} is a homomorphic image of G/N. Therefore Γ/Γ_{1} contains a polycyclic subgroup of finite index. Since N is open and SIN, it follows that Γ_{1} is equicontinuous. Thus Lemma 2.3 applies. □

Following [3] and [24] we call a locally compact group a generalized F C-group if G has a series G = G_{0} ⊇ G_{1} ⊇ … ⊇ G_{n} = {e} of closed normal subgroups such that for every i = 0, 1, …, n − 1, G_{i}/G_{i+1} is a compactly generated group with precompact conjugacy classes. Every compactly generated locally compact group of polynomial growth is a generalized F C group [24, Theorem 2]. Every closed subgroup of a generalized F C group is compactly generated [24, Proposition 2]. A locally compact solvable group G is a generalized F C group if and only if each closed subgroup of G is compactly generated [10, Théorème III.1]. Using Propositions 1 and 7(ii) in [24] it is straighforward to give the following characterization of totally disconnected generalized F C groups:

Proposition 2.8. A totally disconnected locally compact group G is a generalized F C group if and only if it admits a compact open normal subgroup N with the quotient G/N containing a polycyclic subgroup of finite index.

Theorem 2.9. Conditions (i), (ii), and (iii) of Theorem 2.7 are equivalent when G is a totally disconnected generalized F C-group.

Proof. When G is metrizable, this is a special case of Theorem 2.7. We need to show that the implication (ii)⇒(iii) is also true when G is not metrizable.

Note that G is necessarily σ-compact (as it is compactly generated). Let U be a neighbourhood of e contained in the subgroup N of Proposition 2.8. Find a neighbourhood V of e with V^{2} ⊆ U. By [11, Theorem 8.7] V contains a compact normal subgroup K such that G/K is metrizable. Let π: G → G/K denote the canonical homomorphism. Since N is compact, we can use the theorem stating that a factor of a distal flow is distal [5, Corollary 6.10, p. 52] to conclude that the restriction of every inner automorhism of G/K to N/K is distal. As N/K is open, every g ∈ G/K is then distal. Hence, by Theorem 2.7, G/K is SIN. Thus π(V) contains a compact open normal subgroup W. Then Ŵ = π^{−1}(W) ⊆ V K ⊆ U and Ŵ is a compact open normal subgroup of G. □

Since nilpotent groups are distal, Theorem 2.9 implies that a totally disconnected compactly generated locally compact nilpotent group is a SIN group, a result due to Hofmann, Liukkonen, and Mislove [12].

We note that for totally disconnected groups of polynomial growth which are not compactly generated, conditions (i), (ii), (iii) are different. In fact, the equivalence fails already for metabelian groups of polynomial growth. Examples of totally disconnected 2-step nilpotent groups which are not SIN groups can be found in [12] and [31]. An example of a metabelian group of polynomial growth which satisfies (ii) but not (i) (nor (iii)) is also readily available:

Example 2.10. Let H be as in Example 2.5 and let W be the complete wreath product W = ℤ_{2} ≀ H. Evidently, W is not distal but every w ∈ W has finite order, so is distal.

In the remainder of this section we prove that conditions (i), (ii), (iii) of Theorem 2.7 are equivalent for every metrizable compactly generated totally disconnected locally compact metabelian group.

Lemma 2.11. If a locally compact group G contains a normal finitely generated subgroup N and a compact set K such that KN = G, then G is a SIN group.

Proof. Recall that the group of automorphisms of a finitely generated group is countable. Since the centralizer C_{G}(N) of N in G is the kernel of the homomorphism which maps g ∈ G to the restriction of the inner automorphism g · g^{−1} to N, it follows that G/C_{G}(N) is countable. As C_{G}(N) is a closed subgroup and G is of the second category, we conclude that C_{G}(N) is open.

Let U be a neighbourhood of e. Put V = U ∩ C_{G}(N) and let V′ be a neighbourhood of e with g^{−1}V′g ⊆ V for every g ∈ K. Then W = ⋂_{g∈G} gV g^{−1} = ⋂_{g∈K} gV g^{−1} ⊇ V′. Thus W is a neighbourhood of e, invariant under Inn(G) and contained in U. □

Proposition 2.12. If a compactly generated totally disconnected locally compact group G contains a closed cocompact normal SIN subgroup, then G is a SIN group.

Proof. Let N denote the closed cocompact normal SIN subgroup and let a compact open subgroup U of G be given. A routine argument shows that Inn(G) acts equicontinuously on N. Hence, U contains a compact subgroup V of N which is open in N and normal in G.

Let π: G → G/V denote the canonical homomorphism. Since N is cocompact, it is compactly generated [25]. Since V is open in N, π(N) is then finitely generated. It is also normal and there is a compact K ⊆ G/V with Kπ(N) = G/V. Hence, by Lemma 2.11 G/V is a (totally disconnected) SIN group. Thus π(U) contains a compact open normal subgroup W. Then π^{−1}(W) is a compact open normal subgroup contained in UV = U. □

It is well-known that Proposition 2.12 is false for locally compact groups in general (e.g., the motion groups). The following example shows that it can also fail for totally disconnected groups which are not compactly generated:

Example 2.13. Let ℤ^{∗ℕ} = {x ∈ ℤ^{ℕ}; x_{i} ≠ 0 for finitely many i} and give ℤ^{∗ℕ} the discrete topology. Give the multiplicative group {−1, 1}^{ℕ} the product topology. Let φ: {−1, 1}^{ℕ} → Aut(ℤ^{∗ℕ}) be given by φ((ω_{i})_{i=1}^{∞})((x_{i})_{i=1}^{∞}) = (ω_{i}x_{i})_{i=1}^{∞} and let G be the semidirect product G = ℤ^{∗ℕ} ×_{φ} {−1, 1}^{ℕ}.

ℤ^{∗ℕ} × {e} is trivially a closed cocompact normal SIN subgroup of G but G is not a SIN group because for every nonidentity element g = (e, w) ∈ {e} × {−1, 1}^{ℕ} there is a ∈ ℤ^{∗ℕ} with (a, e)(e, w)(a, e)^{−1} ∉ {e} × {−1, 1}^{ℕ}. Indeed, if w_{j} = −1 and a = (δ_{ji})_{i=1}^{∞} then (a, e)(e, w)(a, e)^{−1} = (v, w) where v_{j} = 2.

Lemma 2.14. Let G be a locally compact compactly generated totally disconnected solvable group. Then there exists a closed normal cocompact subgroup N such that \overline{[G, G]} ⊆ N and N/\overline{[G, G]} is topologically isomorphic to ℤ^{d} for some d ≥ 0.

Proof. G/\overline{[G, G]} is a compactly generated totally disconnected abelian group. Hence, it is the direct product AB where A ≅ ℤ^{d} and B is a totally disconnected compact abelian group. Put N = π^{−1}(A) where π: G → G/\overline{[G, G]} is the canonical homomorphism. □

Theorem 2.15. Conditions (i), (ii), and (iii) of Theorem 2.7 are equivalent when G is a metrizable compactly generated totally disconnected locally compact metabelian group.

Proof. Let N be as in Lemma 2.14. To prove the nontrivial implication (ii)⇒(iii) observe that as \overline{[G, G]} is abelian, Theorem 2.7 applies to N. Thus if (ii) holds then N is a SIN group. But then G is a SIN group by Proposition 2.12. □

3. The Choquet–Deny theorem

Let μ be a regular Borel probability measure on a locally compact group G. Recall that G_{μ} denotes the smallest closed subgroup containing the support of μ. μ is called adapted if G_{μ} = G. We will say that μ is a Choquet–Deny measure if every bounded continuous μ-harmonic function is constant on the left cosets of G_{μ}.

We note that in the literature the Choquet–Deny theorem is often understood as the statement that every adapted μ ∈ M_{1}(G) is a Choquet–Deny measure (i.e., all bounded continuous μ-harmonic functions are constant). We emphasize that in this paper the Choquet–Deny theorem is understood as the (formally) stronger statement that every μ ∈ M_{1}(G) is a Choquet–Deny measure. It is not known if the two versions of the Choquet–Deny theorem are equivalent. However, we know of examples of almost connected Lie groups with the property that every adapted spread out probability measure is Choquet–Deny but some nonadapted spread out measures are not. It can be shown (see Lemma 4.1) that the strong version of the theorem is true about G if and only if the weak version holds for every closed subgroup of G.

Throughout the sequel by the weak topology on the set M_{1}(X) of probability measures on a locally compact space X we mean the σ(M_{1}(X), C_{b}(X))-topology where C_{b}(X) is the algebra of bounded continuous functions on X.

Lemma 3.1. (a) If μ is a Choquet–Deny measure on G and N ⊆ G is a closed normal subgroup, then the projection of μ onto G/N is a Choquet–Deny measure on G/N.

(b) If every neighbourhood of e contains a compact normal subgroup N such that the projection of μ onto G/N is a Choquet–Deny measure, then μ is a Choquet–Deny measure.

Proof. We omit a straightforward proof of (a). To prove (b) let us choose, for every neighbourhood Ω of e, a compact normal subgroup N_{Ω} ⊆ Ω such that the projection of μ onto G/N_{Ω} is a Choquet–Deny measure. Denote by π_{Ω}: G → G/N_{Ω} the canonical homomorphism and by ω_{Ω} the normalized Haar measure of N_{Ω}. Directing the neighbourhoods of e by reversed inclusion we obtain a net (ω_{Ω}) in M_{1}(G) which converges weakly to δ_{e}.

Let h be a bounded continuous μ-harmonic function. We need to show that h(xy) = h(x) for every x ∈ G and y ∈ G_{μ}. Now, when Ω is a neighbourhood of e, the function ω_{Ω} ∗ h is a bounded continuous μ-harmonic function constant on the cosets of N_{Ω}. Hence, ω_{Ω} ∗ h = h_{Ω} ∘ π_{Ω} for a bounded continuous function h_{Ω} on G/N_{Ω}. It is clear that h_{Ω} is π_{Ω}μ-harmonic where π_{Ω}μ denotes the projection of μ onto G/N_{Ω}. Moreover, π_{Ω}(G_{μ}) = (G/N_{Ω})_{π_{Ω}μ}. Therefore for x ∈ G and y ∈ G_{μ}, (ω_{Ω} ∗ h)(xy) = h_{Ω}(π_{Ω}(x)π_{Ω}(y)) = h_{Ω}(π_{Ω}(x)) = (ω_{Ω} ∗ h)(x). Since (ω_{Ω} ∗ h)(·) = ∫_{G} h(g^{−1}·) ω_{Ω}(dg) and w-lim_{Ω} ω_{Ω} = δ_{e}, we conclude that h(xy) = h(x). □

Lemma 3.2. Let (μ_{α}) be a net in M_{1}(G). If for every neighbourhood U of e there exists ε ∈ M_{1}(G) such that ε(U) = 1 and the net (μ_{α} ∗ ε) converges weakly, then the net (μ_{α}) converges weakly.

Proof. There exists a compactly supported ν ∈ M_{1}(G) such that the net (μ_{α} ∗ ν) is weakly convergent, and, hence, tight. This implies that the net (μ_{α}) itself is tight. Then by Prohorov’s theorem, every subnet of (μ_{α}) has a weak cluster point.

Therefore it suffices to show that the net (μ_{α}) has a unique cluster point. But if μ′ and μ′′ are cluster points of the net, then, due to our assumption, for every neighbourhood U of e there exists ε ∈ M_{1}(G) such that ε(U) = 1 and μ′ ∗ε = μ′′ ∗ε. As in the proof of Lemma 3.1 we obtain a net (ε_{i}) in M_{1}(G) which converges weakly to δ_{e} and satisfies μ′ ∗ ε_{i} = μ′′ ∗ ε_{i} for every i. Hence, μ′ = μ′′. □

Lemma 3.3. Let G be a totally disconnected locally compact group, τ ∈ Aut(G), and F a finite subset of C(τ). If ν ∈ M_{1}(G) and ν(F) = 1 then the sequence ν ∗ τν ∗ ⋯ ∗ τ^{n−1}ν converges weakly to a probability measure ρ such that ν ∗ τρ = ρ.

Proof. It is clear that if ρ = w-lim_{n→∞} ν ∗ τν ∗ ⋯ ∗ τ^{n−1}ν then ν ∗ τρ = ρ. To see that the limit exists let U be a compact open subgroup. Then there is k ∈ ℕ such that for every n ≥ k, τ^{n}(F) ⊆ U. Let ω_{U} denote the normalized Haar measure of U. Then for n ≥ k, τ^{n}ν ∗ ω_{U} = ω_{U}. Hence, ν ∗ τν ∗ ⋯ ∗ τ^{n−1}ν ∗ ω_{U} converges to ν ∗ τν ∗ ⋯ ∗ τ^{k−1}ν ∗ ω_{U}. By Lemma 3.2, ν ∗ τν ∗ ⋯ ∗ τ^{n−1}ν converges weakly. □

Lemma 3.4. If G is a locally compact group and z ∈ G then C(z) ∩ ⟨z⟩ = {e}.

Proof. Suppose that z^{k} ∈ C(z) for some k > 0. Since C(z) ⊆ C(z^{k}), we obtain z^{k} ∈ C(z^{k}). But when U is a neighbourhood of e in C(z^{k}), then C(z^{k}) = \overline{⋃_{n=1}^{∞} z^{−kn}Uz^{kn}}. This means that C(z^{k}) is either a strange group [22, Definition 1.1], or is compact. Since no locally compact group is strange [22, Theorem 1.8], C(z^{k}) is compact. As z^{k} ∈ C(z^{k}), it follows that C(z) = {e}. □

Suppose that the locally compact group G acts on a locally compact space X so that the mapping G × X ∋ (g, x) → gx ∈ X is continuous. Given ρ ∈ M_{1}(X) and g ∈ G we write gρ for the measure (gρ)(·) = ρ(g^{−1}·). Given μ ∈ M_{1}(G) we denote by μ ∗ ρ the measure (μ ∗ ρ)(·) = ∫_{G}(gρ)(·) μ(dg). Now, if ρ = μ ∗ ρ then for every bounded continuous function f: X → ℂ, the function h(g) = ∫_{X} f(gx) ρ(dx) = ∫_{X} f(x) (gρ)(dx) is a bounded continuous μ-harmonic function. Therefore in order to show that the Choquet–Deny theorem fails for μ it suffices to find g ∈ G_{μ} such that gρ ≠ ρ. This observation is being used in the proof of the next lemma.

Lemma 3.5. Let G be a totally disconnected locally compact group and z an element of G with C(z) ≠ {e}. Let g ∈ C(z) − {e}, and ν = pδ_{g} + (1 − p)δ_{e} where p ∈ (0, 1)−\{\frac{1}{2}\}. Then the Choquet–Deny theorem is false for the measure μ = ν∗δ_{z}.

Proof. Let τ denote the the inner automorphism z · z^{−1}. By Lemma 3.3 the limit ρ = w-lim_{n→∞} ν ∗ τν ∗ ⋯ ∗ τ^{n−1}ν exists and satisfies ν ∗ τρ = ρ. Moreover, ρ(C(z)) = 1.

Note that ⟨z⟩ is necessarily infinite and discrete, so it is a closed subgroup of G (isomorphic to ℤ). Let π: G → G/⟨z⟩ denote the canonical mapping and let ρ̂ = πρ. Then μ ∗ ρ̂ = π(μ ∗ ρ) = π(ν ∗ δ_{z} ∗ ρ) = π(ν ∗ τρ ∗ δ_{z}) = πρ = ρ̂. Since g ∈ G_{μ}, it suffices to show that gρ̂ ≠ ρ̂.

Now, there exists a compact subgroup U of C(z) such that g ∉ U but τ^{j}(g) ∈ U for every j ≥ 1. Let ω_{U} be the normalized Haar measure of U. Then ν∗τν∗⋯∗τ^{n−1}ν∗ω_{U} = ν∗ω_{U} = p(gω_{U})+(1−p)ω_{U}. Thus ρ∗ω_{U} = p(gω_{U})+(1−p)ω_{U} and g(ρ ∗ ω_{U}) = p(g^{2}ω_{U}) + (1 − p)(gω_{U}). Since ρ(C(z)) = 1 and by Lemma 3.4 C(z) ∩ ⟨z⟩ = {e}, we obtain ρ̂(π(U)) = ρ(U⟨z⟩) = ρ(U) = (ρ ∗ ω_{U})(U) = 1 − p and (gρ̂)(π(U)) = (gρ)(U⟨z⟩) = (gρ)(U) = (g(ρ ∗ ω_{U}))(U) = p(g^{2}ω_{U})(U) ≠ 1 − p. □

Theorem 3.6. Let G be a totally disconnected generalized F C-group or a metrizable locally compact compactly generated totally disconnected metabelian group. Then the following conditions are equivalent:

(a) The Choquet–Deny theorem holds for G.

(b) The Choquet–Deny theorem holds for every μ ∈ M_{1}(G) with supp μ of cardinality 2.

(c) G is distal and has polynomial growth.

Proof. (b)⇒(c): We first prove that G is distal. When G is metrizable, this is clear by Lemma 3.5, Theorems 2.9 and 2.15, and Proposition 2.1. Suppose that G is a not necessarily metrizable generalized F C-group. Note that it suffices to show that every neighbourhood of e contains a compact normal subgroup N such that G/N is distal. But as G is compactly generated, given a neighbourhood U of e there exists a compact normal subgroup N ⊆ U such that G/N is metrizable [11, Theorem 8.7]. Every probability measure on G/N with support of cardinality 2 is the canonical image of a similar measure on G. Hence, by Lemma 3.1(a), Condition (b) must hold on G/N and as G/N is a generalized F C-group, it is distal.

We now prove that G is of polynomial growth. Suppose that G is not of polynomial growth. By Proposition 2.8 and Theorem 2.15, G has a compact open normal subgroup N such that the quotient G/N contains a finitely generated solvable subgroup S of finite index (polycyclic when G is an F C-group and metabelian when G is metabelian). By [10, Théorème I.4] S is not of polynomial growth. Hence, by [21, Theorem 3.13 and its proof], S supports a probability measure with a 2element support for which the Choquet–Deny theorem fails. This implies that the Choquet–Deny theorem fails for a similar probability measure on G.

(c)⇒(a): When N is a compact open normal subgroup, G/N is a finitely generated group of polynomial growth, hence, the Choquet–Deny theorem holds for G/N. Since by Theorems 2.9 and 2.15, G has arbitrarily small compact open normal subgroups, Lemma 3.1(b) yields the desired conclusion. □

4. On boundaries of random walks

It is well-known that the bounded μ-harmonic functions can be represented, by means of a “Poisson formula”, as bounded Borel functions on a certain “boundary space”. Let us consider the bounded μ-harmonic functions (on a general locally compact group G) as elements of L^{∞}(G), and let H_{μ} denote the resulting subspace of L^{∞}(G). H_{μ} is invariant under the usual left action of G on L^{∞}(G) and for every absolutely continuous ν ∈ M_{1}(G) and every h ∈ H_{μ}, ν ∗ h is a bounded (left uniformly) continuous μ-harmonic function. When G is locally compact second countable (lcsc), there exists a standard Borel G-space X with a σ-finite quasiinvariant measure α and an equivariant isometry Φ of L^{∞}(X , α) onto H_{μ} [18, §3]. Φ is given by the Poisson formula

(4.1)  (Φf)(g) = ∫_{X} f(gx) ρ(dx)

where ρ is a probability measure on X satisfying μ∗ρ = ρ. The G-space X, called the μ-boundary, or Poisson boundary, is not unique. However, for any two μ-boundaries (X′, α′) and (X′′, α′′), there exists an equivariant isomorphism between L^{∞}(X′, α′) and L^{∞}(X′′, α′′) (which implies that (X′, α′) and (X′′, α′′) are isomorphic up to sets of zero measure). The μ-boundary can be always realized as a topological, compact metric G-space [18, §3].

When the Choquet–Deny theorem holds for μ, the natural realization of the μboundary is the homogeneous space G/G_{μ} where the “Poisson kernel” ρ (cf. Eq. (4.1)) is the point measure δ_{G_{μ}}. When G is a discrete (countable) group then the μ-boundary is a homogeneous space if and only if the Choquet–Deny theorem is true for μ [21, Lemma 1.1 and the remark preceding Proposition 2.6]. The situation is different for continuous groups. When G is an almost connected lcsc group then for every spread out probability measure on G the μ-boundary is a homogeneous space [17, Corollary 4.7]. It is well-known that the μ-boundary of every spread out measure on a connected semisimple Lie group with finite centre is a compact homogeneous space [6, 2]. However, if the μ-boundary of a spread out measure on an amenable lcsc group is a compact homogeneous space, then the Choquet–Deny theorem holds for μ (and the μ-boundary is finite), see [21, Proposition 2.6] and [19, Lemma 2.3], or [2, Propositions IV.8 and IV.7].

Theorem 4.2 which we prove below applies, in particular, to every totally disconnected compactly generated lcsc group of polynomial growth. The result is that for such groups the μ-boundary can be always realized as a homogeneous space, and, as a compact homogeneous space when μ is adapted; when μ is adapted and spread out the μ-boundary is a singleton.

Lemma 4.1. A probability measure μ on a locally compact group G is a Choquet– Deny measure if and only if the restriction of μ to G_{μ} is a Choquet–Deny measure (on G_{μ}).

Proof. Let μ′ denote the restriction of μ to G_{μ}. The restriction of a μ-harmonic function to G_{μ} is μ′-harmonic; moreover, if h is μ-harmonic then for every g ∈ G the left translate (gh)(·) = h(g^{−1}·) is also μ-harmonic. Hence, if μ′ is Choquet–Deny then so is μ. The converse is equally obvious when G_{μ} is open, because then every bounded continuous μ′-harmonic function trivially extends to a bounded continuous μ-harmonic function. However, in general, a technical argument is called for.

Let us first consider the case that G is second countable. Let h′ be a bounded continuous μ′-harmonic function. As G is second countable, the canonical projection π: G → G/G_{μ} admits a Borel cross-section κ. Since for every g ∈ G, κ(π(g))^{−1}g ∈ G_{μ}, we can define a function h: G → ℂ by h(g) = h′(κ(π(g))^{−1}g). h is a bounded (in general, discontinuous) μ-harmonic function.

Let (ε_{n}) be a sequence of absolutely continuous probability measures on G converging weakly to δ_{e}. Then the sequence (ε_{n} ∗ h) converges in the weak* topology of L^{∞}(G) to h. Since ε_{n} ∗ h is a bounded continuous μ-harmonic function and μ is a Choquet–Deny measure, it follows that there exists a bounded Borel function ĥ: G/G_{μ} → ℂ such that h = ĥ ∘ π λ-a.e., where λ is the Haar measure of G.

Now, the mapping φ: (G/G_{μ}) × G_{μ} → G given by φ(x, g) = κ(x)g is a Borel isomorphism. Moreover, if ν is a σ-finite quasiinvariant measure on G/G_{μ} and λ′ the Haar measure of G_{μ}, then the measure φ(ν × λ′) = (ν × λ′) ∘ φ^{−1} is equivalent to the Haar measure λ of G. Consequently,

0 =  ∫_{(G/G_{μ})×G_{μ}} |h′ ∘ φ − ĥ ∘ π ∘ φ| d(ν × λ′)

= ∫_{G/G_{μ}} [ ∫_{G_{μ}} |(h ∘ φ)(x, g) − (ĥ ∘ π ∘ φ)(x, g)| λ′(dg)] ν(dx)

= ∫_{G/G_{μ}} [ ∫_{G_{μ}} |h′(g) − ĥ(x)| λ′(dg)] ν(dx).

Thus for ν-a.e. x ∈ G/G_{μ}, ∫_{G_{μ}} |h′(g) − ĥ(x)| λ′(dg) = 0. Hence, as h′ is continuous, it is constant.

Consider now the general case that G is not necessarily second countable. Observe that due to the regularity of μ and local compactness of G, G_{μ} is σ-compact and, hence, there is also an open σ-compact subgroup G_{1} with μ(G_{1}) = 1. Since G_{1} is open it is clear that the restriction of μ to G_{1} is a Choquet–Deny measure. Hence, we may assume that G itself is σ-compact. By Lemma 3.1(b) it suffices to show that every neighbourhood U of e in G_{μ} contains a compact normal subgroup N such that the projection of μ′ onto G_{μ}/N is Choquet–Deny. But by [11, Theorem 8.7] there exists a compact normal subgroup K of G such that K ∩ G_{μ} ⊆ U and G/K is second countable. Let π_{K}: G → G/K denote the canonical homomorphism. Since (G/K)_{π_{K}μ} = π_{K}(G_{μ}), combining Lemma 3.1(a) with what we just proved for second countable groups, we conclude that the restriction of π_{K}μ to π_{K}(G_{μ}) is a Choquet–Deny measure. As π_{K}(G_{μ}) is canonically isomorphic to G_{μ}/(K ∩ G_{μ}), it follows that the projection of μ′ onto G_{μ}/(K ∩G_{μ}) is a Choquet–Deny measure. □

Theorem 4.2. Let μ ∈ M_{1}(G) where G is a lcsc group. If G contains a compact normal subgroup K such that the projection of μ onto G/K is a Choquet–Deny measure, then the μ-boundary can be realized as a homogeneous space; when μ is adapted, the μ-boundary can be realized as a compact homogeneous space on which K acts transitively.

Proof. Denote by π: G → G/K the canonical homomorphism.

Suppose that μ is adapted and let (X , α) be the μ-boundary realized as a standard Borel G-space. Let f ∈ L^{∞}(X , α) be invariant under the action of K. Then the corresponding μ-harmonic function h = Φf ∈ H_{μ} (cf. Eq. (4.1)) is also invariant under the (left) action of K. Hence, h = ĥ ∘ π where ĥ ∈ H_{πμ}. Since πμ is adapted and the Choquet–Deny theorem holds on G/K, it follows that h is constant. Thus so is f. As X is a standard Borel G-space this implies that K acts ergodically on X, and, hence, α is carried on an orbit of K [33, Corollary 2.1.21 and Proposition 2.1.10]. Consequently, the μ-boundary can be realized as a compact homogeneous space of G on which K acts transitively.

When μ is not necessarily adapted, let μ′ denote the restriction of μ to G_{μ} and let (X′, α′) be a realization of the μ′-boundary as a standard Borel G_{μ}-space. By Lemma 4.1 the restriction of πμ to π(G_{μ}) = (G/K)_{πμ} is a Choquet–Deny measure. Since G_{μ}/(G_{μ} ∩ K) ≅ π(G_{μ}), it follows that the projection of μ′ onto G_{μ}/(G_{μ} ∩ K) is a Choquet–Deny measure. As μ′ is adapted, we may assume that X′ is a homogeneous space of G_{μ} (on which G_{μ} ∩ K acts transitively). Now, by [20, Proposition 3.5 and Remark 3.9] the μ-boundary can be realized as the skew product X = G/G_{μ} ×_{γ} X′ (the G-space induced from the G_{μ}-space X′ [33, p. 75]), where γ: G × G/G_{μ} → G_{μ} is the cocycle associated with a Borel cross section of the canonical projection of G on G/G_{μ}. It follows that G acts transitively on X. This means that the μ-boundary can be realized as a homogeneous space of G. □

Corollary 4.3. Let G be a totally disconnected compactly generated lcsc group of polynomial growth. Then for every μ ∈ M_{1}(G) the μ-boundary can be realized as a homogeneous space of G; when μ is adapted, the μ-boundary can be realized as a compact homogeneous space.

The next corollary can be regarded as a generalization of the implication (c)⇒(a) of Theorem 3.6. Contrary to the proof of Theorem 3.6, the proof of Corollary 4.4 does not rely on equicontinuity of Inn(G).

Corollary 4.4. Let G be a locally compact group containing a compact normal subgroup K such that the Choquet–Deny theorem holds for G/K and Inn(G) acts distally on K. Then the Choquet–Deny theorem holds for G.

Proof. It is not difficult to see that if a locally compact group G contains a compact normal subgroup K such that the Choquet–Deny theorem holds for G/K and Inn(G) acts distally on K, then the same is true for every closed subgroup and every quotient of G. Let μ ∈ M_{1}(G). To show that μ is a Choquet–Deny measure it suffices to show that the restriction, μ′, of μ to G_{μ} is Choquet–Deny. By Lemma 3.1(b), to show the latter it is enough to show that every neighbourhood of e in G_{μ} contains a compact normal subgroup N such that the projection of μ′ onto G_{μ}/N is Choquet–Deny. But as G_{μ} is σ-compact, every neighbourhood of e in G_{μ} contains a compact normal subgroup with second countable quotient. Hence, it is enough to prove that if a lcsc group G contains a compact normal subgroup K such that the Choquet–Deny theorem holds for G/K and Inn(G) acts distally on K, then every adapted probability measure on G is a Choquet–Deny measure.

For such G and μ, by Theorem 4.2, the μ-boundary has the form G/H where K acts transitively on G/H, i.e., G = KH. Let ρ denote the Poisson kernel. Note that due to the identity ρ = μ ∗ ρ and adaptedness of μ, it suffices to show that ρ is a point measure (this will imply that G/H is a singleton).

Now, by [14, Proposition 2.8] there exists a sequence (h_{n}) in G such that the sequence (h_{n}ρ) converges weakly to a point measure δ_{x_{0}}. Since G = KH and K is compact, we may assume that h_{n} ∈ H for all n. Next, by [22, Lemma 2.8] we may assume that there is a Borel set B ⊆ G/H such that ρ(B) = 1 and lim_{n→∞} h_{n}x = x_{0} for every x ∈ B. It is enough to show that B is a singleton.

Consider the compact homogeneous space K/(K ∩ H). The formulas h^{•}k = hkh^{−1} and h_{•}k(K ∩ H) = hkh^{−1}(K ∩ H), k ∈ K, define actions of H on K and K/(K ∩ H), respectively. Clearly, the _{•} -action is a factor of the ^{•} -action. As ^{•} is distal, so is _{•} [5, Corollary 6.10, p. 52].

Let x_{1}, x_{2} ∈ B. Write x_{1} = k_{1}H and x_{2} = k_{2}H with k_{1}, k_{2} ∈ K. Then lim_{n→∞} h_{n}x_{j} = lim_{n→∞} h_{n}k_{j}h_{n}^{−1}H = x_{0} for j = 1, 2. Since the compact homogeneous spaces G/H and K/(K ∩ H) are isomorphic as K-spaces, we then obtain lim_{n→∞} h_{n}_{•}k_{1}(K ∩ H) = lim_{n→∞} h_{n}_{•}k_{2}(K ∩ H). Since _{•} is distal, k_{1}(K ∩ H) = k_{2}(K ∩ H) and, hence, x_{1} = x_{2}. Therefore B is a singleton. □

Example 4.5. Recall that G is called an IN group if it contains a compact neighbourhood of e, invariant under Inn(G). By [8, Theorem 2.5] every locally compact IN group G contains a compact normal subgroup N such that G/N is a SIN group. When G is almost connected, it follows from Iwasawa’s theorem on automorphisms of compact groups [13, Theorem 1] that the natural image of Inn(G) in Aut(N) is compact (in the usual topology). Hence, the Ascoli theorem for automorphism groups [9, Theorem 4.1] yields that Inn(G) acts equicontinuously on N. But by [14, Corollary 6.5] and [8, Theorem 2.9] the Choquet–Deny theorem holds for every almost connected locally compact SIN group (see also [14, Corollary 6.6]). Thus Corollary 4.4 yields that the Choquet–Deny theorem is true for every locally compact almost connected IN group.

Example 4.6. Let τ be the automorphism of the torus 𝕋^{3}, defined by τ(x, y, z) = (x, xy, xyz). Then τ is distal but not equicontinuous. By Corollary 4.4 the Choquet– Deny theorem is true for the 3-step nilpotent group 𝕋^{3} ×_{τ} ℤ.

Example 4.7. Let τ be the shift τ((x_{i})_{i∈ℤ}) = (x_{i+1})_{i∈ℤ} on the compact abelian group ℤ_{2}^{ℤ}, and let G = ℤ_{2}^{ℤ}×_{τ} ℤ. Since C(τ) = {x ∈ ℤ_{2}^{ℤ}; there exists k ∈ ℤ with x_{i} = 0 for every i ≥ k}, Inn(G) does not act distally on ℤ_{2}^{ℤ} × {0}. By Theorem 3.6 the Choquet–Deny theorem is not true for G.

Let μ ∈ M_{1}(G) be adapted. According to Theorem 4.2 the μ-boundary has the form G/H where G = (ℤ_{2}^{ℤ} × {0})H. It is not difficult to see that a closed subgroup H ⊆ G satisfies G = (ℤ_{2}^{ℤ} × {0})H if and only if there is a closed τ-invariant subgroup T ⊆ ℤ_{2}^{ℤ} and g ∈ G such that gHg^{−1} = T × ℤ. We may therefore assume that H = T × ℤ where T is a closed τ-invariant subgroup. Now, the formula (x, y)(zT) = xτ^{y}(z)T, x, z ∈ ℤ_{2}^{ℤ}, y ∈ ℤ, defines an action of G on ℤ_{2}^{ℤ}/T under which ℤ_{2}^{ℤ}/T becomes a homogeneous space of G, isomorphic to G/H. Thus for an adapted μ ∈ M_{1}(G), the μ-boundary can be realized as one of the G-spaces ℤ_{2}^{ℤ}/T, where T is a closed τ-invariant subgroup of ℤ_{2}^{ℤ}.

Let for k = 1, 2, …, S_{k} = {x ∈ ℤ_{2}^{ℤ} τ^{k}(x) = x}. Then S_{k} is a closed τ-invariant subgroup of ℤ_{2}^{ℤ} and it can be shown that T is a closed τ-invariant subgroup of ℤ_{2}^{ℤ} if and only if T = ℤ_{2}^{ℤ} or T is a τ-invariant subgroup of S_{k} for some k. In particular, proper τ-invariant subgroups are finite. Let T denote the class of closed τ-invariant subgroups of ℤ_{2}^{ℤ}. The G-spaces ℤ_{2}^{ℤ}/T, T ∈ T are mutually nonisomorphic and each of them is an equivariant image of ℤ_{2}^{ℤ} = ℤ_{2}^{ℤ}/{e}.

One can construct a family μ_{T}, T ∈ T , of discrete probability measures on G such that ℤ_{2}^{ℤ}/T is the μ_{T} -boundary for every T ∈ T . We refrain from going into the details here as this would require a longer digression into the theory of the μ-boundaries. A more difficult question concerns determining, when μ ∈ M_{1}(G) is given, which of the spaces ℤ_{2}^{ℤ}/T is the μ-boundary. In particular, one would like to know for which μ ∈ M_{1}(G) the μ-boundary is a singleton. In addition to the case of spread out measures, this is so for every adapted probability measure which induces a recurrent random walk on ℤ ≅ G/(ℤ_{2}^{ℤ} × {0}). We do not know of any relevant conditions that are both sufficient and necessary.

Acknowledgement. The second author thanks the School of Mathematics and Statistics at Carleton University for its hospitality.

Note added in proof. Since this paper was completed the first named author proved that the results of Baumgartner and Willis [4] remain true for nonmetrizable

The Choquet–Deny theorem 173

groups. As a consequence, Proposition 2.1, Lemmas 2.2 and 2.3, Corollary 2.4, and Theorems 2.7, 2.15, and 3.6 of the present work remain true without the assumption that G be metrizable.

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School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada K1S 5B6 wjaworsk@math.carleton.ca

Indian Statistical Institute, Statistics and Mathematics Unit, 8th Mile Mysore Road, Bangalore 560 059, India creraja@isibang.ac.in

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