EXPANSIVE ALGEBRAIC ACTIONS OF DISCRETE RESIDUALLY FINITE AMENABLE GROUPS AND THEIR ENTROPY

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1 EXPANSIVE ALGEBRAIC ACTIONS OF DISCRETE RESIDUALLY FINITE AMENABLE GROUPS AND THEIR ENTROPY CHRISTOPHER DENINGER AND KLAUS SCHMIDT Abstract. We prove an entropy ormula or certain expansive actions o a countable discrete residually inite group Γ by automorphisms o compact abelian groups in terms o Fuglede-Kadison determinants. This extends an earlier result proved by the irst author under somewhat more restrictive conditions. The main tools or this generalization are a representation o the Γ- action by means o a undamental homoclinic point and the description o entropy in terms o the renormalized logarithmic growth-rate o the set o Γ n-ixed points, where (Γ n, n 1) is a decreasing sequence o inite index normal subgroups o Γ with trivial intersection. 1. Introduction For a countable discrete amenable group Γ and an element = γ γ in the integral group ring ZΓ consider the quotient ZΓ/ZΓ o ZΓ by the let ideal ZΓ generated by. It is a discrete abelian group with a let Γ-action by multiplication. The Pontryagin dual (1.1) X = ZΓ/ZΓ is a compact abelian group with a let action o Γ by continuous group automorphisms. More explicitly, X is the ollowing closed subshit o the ull shit (R/Z) Γ = ẐΓ with Γ-action given by (γ x) γ = x γ 1 γ. The elements o X are the sequences (x γ ) in (R/Z) Γ which satisy the equations γ x γγ = 0 in R/Z or all γ in Γ. γ For Γ = Z d the theory o such dynamical systems has been extensively studied, see [13] or example. For general Γ the investigation o such systems was initiated in [6] Mathematics Subject Classiication. 37A35, 37A45, 37B05, 37B05, 37C85. Key words and phrases. Algebraic dynamical systems, entropy o countable group actions, Fuglede-Kadison determinants. The second author (KS) was partially supported by the FWF Research Grant P N05. 1

2 2 CHRISTOPHER DENINGER AND KLAUS SCHMIDT The various deinitions o entropy, topological, metric and measure theoretic with respect to Haar measure, all coincide or this action, see e.g. [3, section 2]. We denote by h this common entropy. I is a unit in the convolution algebra L 1 (Γ), then the action o Γ on X is expansive. I in addition is positive in the von Neumann algebra N Γ and Γ has a log-strong Følner sequence, it was shown in [3] that h = log det N Γ, where det N Γ is the Fuglede Kadison determinant, [7], [11]. Currently the only groups known to have a log-strong Følner sequence are the virtually nilpotent ones, so this condition is unortunate, even though it is only needed or the inequality h log det N Γ in [3]. Recall that a countable group Γ is residually inite i there is a sequence Γ n o normal subgroups o inite index whose intersection is trivial. Our main result is this: Theorem 1.1. Let Γ be a countable discrete amenable and residually inite group and an element o ZΓ. Then the action o Γ on X is expansive i and only i is a unit in L 1 (Γ). In this case we have the ormula (1.2) h = log det N Γ. For example, Γ could be any initely generated solvable subgroup o a matrix group over a ield. I = γ γ is invertible in L 1 (Γ) then det N Γ = det N Γ, where = γ γ 1, since the Fuglede-Kadison determinant o an invertible operator is equal to that o its adjoint. From (1.2) we thereore obtain that h = h. This question had been let open in [3] even or the integral Heisenberg group. Dynamically the proo o ormula (1.2) is based on a description o the entropy or expansive Γ-actions as a renormalized logarithmic growth rate o the number o Γ n -ixed points, where (Γ n, n 1) is a decreasing sequence o inite index normal subgroups with trivial intersection. In the case o expansive Z d -actions this relation between entropy and periodic points is known and the methods can be carried over to our case. It remains to prove that det N Γ is the limit o certain inite dimensional determinants. This is a well-known problem in the context o L 2 -invariants or residually inite groups c.. [11, Chapter 13]. There one tries to approximate L 2 - Betti numbers, L 2 -signatures etc. o coverings with covering group Γ by the corresponding quantities or the inite groups Γ/Γ n. For the combinatorial L 2 -torsion which involves Fuglede Kadison determinants there is no general result in this direction however.

3 EXPANSIVE ALGEBRAIC ACTIONS 3 Nonetheless in our situation which is quite avourable analytically it is not diicult to prove the required approximation property o det N Γ directly. It is a pleasure or us to thank our colleagues Gabor Elek, Doug Lind and Wolgang Lück or helpul comments. We are also grateul to Doug Lind or alerting us to the reerence [14]. The irst author would like to thank the Erwin Schrödinger Institute or support. 2. Group rings Let Γ be a countable discrete group with identity element 1 = 1 Γ and integral group ring ZΓ. We denote by L (Γ) the set o all bounded maps rom Γ to R, write every w L (Γ) as (w γ ) with w γ R or every γ Γ, and denote by w = sup γ Γ w γ the supremum norm on L (Γ). For p [1, ) we set L p (Γ) = { ( w L (Γ) : w p := γ Γ For γ Γ we deine e(γ) L 1 (Γ) L (Γ) by { 1 i γ = γ (2.1) e(γ) γ = 0 otherwise. ) 1/p } w γ p <. Every h in L 1 (Γ) can be uniquely written as a convergent series (2.2) h = γ Γ h γ e(γ) with h γ R and γ Γ h γ <. In this notation multiplication in the convolution algebra L 1 (Γ) takes the orm h h = ( h γγ 1h γ )e(γ) (2.3) h γ h γ e(γγ ) = γ,γ Γ γ Γ γ Γ = ( ) h γ h e(γ). γ 1 γ γ Γ The involution h h in L 1 (Γ) is deined by (2.4) h = γ Γ γ Γ h γ 1e(γ) = γ Γ h γ e(γ 1 ) and satisies that (g h) = h g or g, h L 1 (Γ). An element h L 1 (Γ) is sel-adjoint i h = h. Note that the integral group ring ZΓ o inite ormal sums γ a γγ with a γ Z can be viewed as a subring o L 1 (Γ) by identiying γ a γγ with γ a γe(γ).

4 4 CHRISTOPHER DENINGER AND KLAUS SCHMIDT Multiplication in L 1 (Γ) can be extended urther to commuting (let) actions (h, w) λ h w and (h, w) ρ h w o L 1 (Γ) on L (Γ) by setting (2.5) (λ h w) γ = (h w) γ = h γγ 1w γ = h γ w γ 1 γ, γ Γ γ Γ (ρ h w) γ = (w h ) γ = w γγ 1h γ = w γγ h γ γ Γ γ Γ or every h L 1 (Γ), w L (Γ) and γ Γ. The let and right shit actions λ and ρ o Γ on L (Γ) are given by or, equivalently, by λ γ w = λ e(γ) w, ρ γ w = ρ e(γ) w (2.6) (λ γ w) γ = w γ 1 γ, (ργ w) γ = w γ γ or every w L (Γ) and γ, γ Γ. For the ollowing lemmas we ix h L 1 (Γ) and deine the linear operators ρ h, ρ h : L (Γ) L (Γ) by (2.4) (2.5). For v L p (Γ) and w L q (Γ) with 1 p + 1 q = 1, 1 p, q, we set (2.7) v, w = γ Γ v γ w γ. For 1 < p this pairing identiies L p (Γ) with the dual L q (Γ) as Banach spaces and L p (Γ) acquires a weak -topology. In this topology (v n ) converges to v i and only i lim n v n, w = v, w or all w in L q (Γ). Note that (2.8) v, ρ h w = v, w h = v h, w, v, λ h w = v, h w = h v, w or every h L 1 (Γ). Finally we put, or every h L 1 (Γ) and 1 p, (2.9) K p (h) = {g L p (Γ) : ρ h g = 0}, V p (h) = ρ h (L p (Γ)). 3. Algebraic Γ-actions Deinition 3.1. Let Γ be a countable discrete group. An algebraic Γ-action is a homomorphism α: γ α γ rom Γ into the group Aut(X) o continuous automorphisms o a compact abelian group X. I α is an algebraic Γ-action on a compact abelian group X then α is expansive i there exists an open neighbourhood U o the identity in X with γ Γ αγ (U) = 0. In this section we consider algebraic Γ-actions o a very special nature. Fix a countable discrete group Γ and consider the compact abelian group X = T Γ consisting o all maps rom Γ to T = R/Z under point-wise addition. Each

5 EXPANSIVE ALGEBRAIC ACTIONS 5 x X is written as (x γ ) = (x γ, γ Γ), where x γ T denotes the value o x at γ Γ. We identiy the Pontryagin dual X o X with the group ring ZΓ under the pairing (3.1), x = e 2πi P γ Γ γxγ, where = γ Γ γγ ZΓ and x = (x γ ) X. The let and right shit actions λ and ρ o Γ on X are deined by (3.2) (λ γ x) γ = x γ 1 γ and (ργ x) γ = x γ γ or every γ, γ Γ and x X. As in (2.5) we can extend these actions o Γ to commuting (let) actions λ and ρ o ZΓ on X by setting (3.3) λ x = γ Γ γ λ γ x and ρ x = γ Γ γ ρ γ x or ZΓ and x X (c. (2.4)). For every ZΓ, the maps λ, ρ : X X are continuous group homomorphisms which are dual to let and right multiplication by and, respectively, on X = ZΓ. For the ollowing discussion we set (3.4) L (Γ, Z) = {w = (w γ ) L (Γ) : w γ Z or every γ Γ}. We ix ZΓ, set (3.5) X = ker(ρ ) = {x X : ρ x = 0} = ZΓ/ZΓ, and denote by (3.6) α = λ X the restriction to X o the Γ-action λ on X. Since α γ Aut(X ) or every γ Γ, α is an algebraic Γ-action on the compact abelian group X. The map η : L (Γ) X, given by (3.7) η(w) γ = w γ (mod 1) or every w = (w γ ) L (Γ) and γ Γ, is a continuous surjective group homomorphism with (3.8) η λ γ = λ γ η, η ρ γ = ρ γ η or every γ Γ, and the linearization (3.9) W = η 1 (X ) = ρ 1 (L (Γ, Z)) = {w L (Γ) : ρ w L (Γ, Z)} o X is a weak -closed and λ-invariant subgroup with ker(η) = L (Γ, Z) W. Theorem 3.2. Let Γ be a countable group, ZΓ, and let α be the algebraic Γ-action on X deined in (3.5) (3.6). The ollowing conditions are equivalent. (1) The action α is expansive;

6 6 CHRISTOPHER DENINGER AND KLAUS SCHMIDT (2) K () = {0} (c. (2.5) and (2.9)); (3) is invertible in L 1 (Γ). Proo. This ollows by combining [6, Theorem 8.1] with [2, Theorem 1]. For the convenience o the reader we include the argument. Let d be the usual metric (3.10) d(s 1, s 2 ) = min { s 1 s 2 : s i R and s i = s i (mod 1) or i = 1, 2} on T. I there exists a nonzero element v K () then η(cv) X or every c R, and by choosing c suiciently small we can ind, or every ε > 0, a nonzero element x (ε) X with d(x (ε) γ, 0) < ε or every γ Γ. This proves that α is nonexpansive. Conversely, i α is nonexpansive, then we can ind a nonzero element x X with d(x γ, 0) < (3 1 ) 1 or every γ Γ. We choose x η 1 ({x}) W = η 1 (X ) with x γ < (3 1 ) 1 or every γ Γ. By deinition o X, ρ x L (Γ, Z), and the smallness o the coordinates o x implies that x K (). This proves that (1) (2). I K () = {0} then V 1 ( ) is dense in L 1 (Γ) by (2.8) and the Hahn-Banach theorem. Since the group o units in L 1 (Γ) is open, V 1 ( ) contains a unit. Hence there exists a g L 1 (Γ) with g = 1. By [8, p. 122], is invertible in L 1 (Γ), which proves (3). The implication (3) (2) is obvious. 4. Homoclinic points Deinition 4.1. Let α be an algebraic action o a countable discrete group Γ on a compact abelian group X with identity element 0. A point x X is α-homoclinic (or simply homoclinic) i lim γ α γ x = 0, i.e. i or every neighbourhood U o 0 in X there is a inite subset F o Γ with α γ x U or all γ Γ F. The set α (X) o all α-homoclinic points in X is a subgroup o X, called the homoclinic group o α. Following [9] we call an α-homoclinic point x X undamental i the homoclinic group α (X) is generated by the orbit {α γ x : γ Γ} o x. Proposition 4.2. Let Γ be a countable group and ZΓ an element which is invertible in L 1 (Γ). I w = 1 L 1 (Γ) and ξ = η ρ w : L (Γ, Z) X (c. (2.5) and (3.5) (3.7)), then ξ is a surjective group homomorphism with the ollowing properties. (1) ker(ξ) = ρ (L (Γ, Z));

7 EXPANSIVE ALGEBRAIC ACTIONS 7 (2) ξ λ γ = α γ ξ or every γ Γ; (3) ξ is continuous in the weak -topology on closed, bounded subsets o L (Γ, Z). Proo. We set w = ( ) 1 = ( 1 ). By deinition, ρ w = w = e(1) and hence w W and x = η( w) X (c. (3.9)). Since w L 1 (Γ), x α (X ). The λ-invariance o W implies that ρ w h = λ h w W or every h ZΓ. Since W is weak -closed and ρ w is weak -continuous on bounded subsets o L (Γ, Z), it ollows that ρ w (L (Γ, Z)) W. In order to prove that ρ w (L (Γ, Z)) = W we ix w W, set v = ρ w L (Γ, Z), and obtain that w = ρ w v. The group homomorphism (4.1) ξ = η ρ w : L (Γ, Z) X is thus surjective, and the equivariance o ξ is obvious. I B L (Γ, Z) is a closed, bounded subset, then the weak -topology coincides with the topology o coordinate-wise convergence, and ξ is obviously continuous in that topology. Theorem 4.3. Let Γ be a countable residually inite discrete group and ZΓ. I the algebraic Γ-action α on the compact abelian group X in (3.5) (3.6) is expansive, and i w = 1 L 1 (Γ) is the inverse o in L 1 (Γ) described in Theorem 3.2, then x = η( (w ) ) X is a undamental homoclinic point o α. Proo. Since w = (w ) L 1 (Γ), x α (X ). I x α (X ) is an arbitrary α -homoclinic point, then we can ind a w η 1 ({x}) with lim γ w γ = 0 and hence with v = ρ (w) L 1 (Γ, Z) = L 1 (Γ) L (Γ, Z). The point w = ρ w v lies in the group generated by the λ-orbit {λ γ w : γ Γ} o w, and by applying η we see that x lies in the group generated by the α -orbit o x. This proves that x is undamental. Theorem 4.4 (Speciication Theorem). Let Γ be a countable residually inite group and ZΓ an element such that the algebraic Γ-action α on the compact abelian group X in (3.5) (3.6) is expansive. Then there exists, or every ε > 0, a inite subset F ε Γ with the ollowing property: i C 1, C 2 are subsets o Γ with F ε C 1 F ε C 2 =, then we can ind, or every pair o points x (1), x (2) X, a point y X with d(x (i) γ, y γ ) < ε or every γ C i, i = 1, 2 (c. (3.10)).

8 8 CHRISTOPHER DENINGER AND KLAUS SCHMIDT For the proo o Theorem 4.4 we need an elementary lemma. Lemma 4.5. For every x X there exists an element v L (Γ, Z) with ξ(v) = x and v 1 /2. Proo. Choose w W = η 1 (X ) L (Γ) with η(w) = x and 1/2 w γ < 1/2 or every γ Γ. Then v = ρ w L (Γ, Z), v 1 /2 and ξ(v) = x by Proposition 4.2. Proo o Theorem 4.4. Consider the point w = 1 L 1 (Γ) described in Theorem 3.2. We can ind, or every ε > 0, a inite set F ε Γ with γ Γ F ε (w ) γ < ε/ 1. Put { (w w γ = ) γ i γ F ε, 0 otherwise and note that (ρ w y) γ (ρ w y) γ < ε/2 or every γ Γ and every y L (Γ, Z) with y 1 /2. Lemma 4.5 allows us to ind points v (i) L (Γ, Z) such that ξ(v (i) ) = x (i) and v (i) 1 /2 or i = 1, 2. Let v L (Γ, Z) be a point with v 1 /2 and v γ = v γ (i) Then d(x (i) or γ C i Fε 1, i = 1, 2, where Fε 1 = {γ 1 : γ F ε }. γ, η(ρ w v (i) ) γ ) < ε/2 or every γ Γ and (ρ w v (i) ) γ = (ρ w v) γ or every γ C i, i = 1, 2. By setting y = ξ(v) we obtain that d(x γ (i), y γ ) < ε or every γ C i, i = 1, 2. This proves our claim. Theorem 4.6. Let Γ be a countable residually inite group and ZΓ an element such that the algebraic Γ-action α on the compact abelian group X in (3.5) (3.6) is expansive. Then the homoclinic group α (X ) is countable and dense in X. Proo. The countability o α (X ) is proved exactly as in [9, Lemma 3.2]. In order to veriy that α (X ) is dense in X we consider the continuous surjective group homomorphism ξ : L (Γ, Z) X in Proposition 4.2. We set B = {v L (Γ, Z) : v 1 /2} and note that ξ(b) = X by Lemma 4.5, and that the restriction o ξ to B is continuous in the weak -topology. Since B L 1 (Γ, Z) is countable and weak -dense in B, the countable set ξ(b L 1 (Γ, Z)) α (X ) is dense in X. We end this section with a lemma which implies that the point w appearing in the Theorems 4.3 and 4.4 decays rapidly. L1 (Γ) Proposition 4.7. Let Γ be a countable discrete group and ZΓ. I α is expansive there exists, or every L > 0 and ε > 0, a inite subset F (L, ε) Γ with the ollowing property: i w W satisies that w L and (ρ w) γ = 0 or every γ F (L, ε), then w 1 < ε.

9 EXPANSIVE ALGEBRAIC ACTIONS 9 I the group Γ is residually inite, initely generated and has polynomial growth, then the point w has exponential decay in the word metric on Γ. Proo. We argue by contradiction and assume that there exist an ε > 0, an increasing sequence (F n ) o inite subsets in Γ with n 1 F n = Γ and a sequence (w (n), n 1) in W such that, or every n 1, w (n) L, w (n) 1 > ε and ρ w γ (n) = 0 or every γ F n. I w (0) is the limit o a weak -convergent subsequence o (w (n) ), then w (0) 1 ε and w(0) ker(ρ ), contrary to our hypothesis that α is expansive and ker(ρ ) is thereore equal to 0. Now suppose that Γ is residually inite, initely generated and has polynomial growth. We choose and ix a inite symmetric set o generators F o Γ and write δ F or the word-metric on Γ. For every γ Γ we denote by l F (γ) the length o the shortest expression o γ Γ in terms o elements o F, where l F (1) = 0. Since Γ has polynomial growth there exist constants c, M 1 such that the set (4.2) B F (n) = {γ Γ : l F (γ) n} has cardinality cm n or every n 0. The point w = (w ) L 1 (Γ) satisies that ρ ω = e(1). We set L = w and use the irst part o this proposition to ind an R 1 such that w 1 < L/2 or every w W with w L and ρ w 0 on B F (R). Our choice o R guarantees that w γ < L/2 or every γ B F (2R) B F (R) and, by induction, that w γ < L/2 k or every k 1 and every γ B F (kr) B F ((k 1)R). This proves exponential decay o the coordinates o w (and hence o w ) in the word metric on Γ. 5. Entropy and periodic points Let Γ be a countable discrete group and K Γ a inite set. A inite set Q Γ is let (K, ε)-invariant i γq Q / Q < ε, γ K and right (K, ε)-invariant i Qγ Q / Q < ε. γ K I Q satisies both these conditions it is (K, ε)-invariant. A sequence (Q n, n 1) o inite subsets o Γ is a let Følner sequence i there exists, or every inite subset K Γ and every ε > 0, an N 1 such

10 10 CHRISTOPHER DENINGER AND KLAUS SCHMIDT that Q n is let (K, ε)-invariant or every n N. The deinitions o right and two-sided Følner sequences are analogous. The group Γ is amenable i it has a let Følner sequence. I Γ is amenable it also has right and two-sided Følner sequences. Let Γ be a countable residually inite discrete group. I (Γ n, n 1) is a sequence o inite index normal subgroups in Γ we say that (5.1) lim n Γ n = {1} i we can ind, or every inite set K Γ, an N 1 with Γ n (K 1 K) = {1} or every n N. Clearly, such sequences exist. Theorem 5.1. Let Γ be a countable residually inite amenable group, ZΓ, and let α be the Γ-action on X deined in (3.5) (3.6). I X 0 and α is expansive then h(α ) > 0. Proo. Let (Γ n, n 1) be a decreasing sequence o inite index normal subgroups o Γ with n 1 Γ n = {1}. For notational simplicity set w = (w ) (c. Theorem 4.3). The homoclinic group α (X ) is dense in X by Theorem 4.6 and, since X 0 by assumption, the undamental homoclinic point x = x = η( w) is nonzero. Hence there exist a γ 0 Γ and a inite subset F Γ with w γ0 / Z and γ Γ F w γ < d( x γ0, 0)/2, where d(, ) is the metric on T deined in (3.10)). We choose n suiciently large so that the sets γf, γ Γ n, are all disjoint. For every ω Ω = {0, 1} Γn we deine a point x (ω) X by setting w (ω) = γ Γ n ω γ λ γ w and x (ω) = η(w (ω) ) = γ Γ n ω γ λ γ x. We ix γ Γ n or the moment and consider two points ω, ω ω γ ω γ. Then d(x (ω) γγ 0, x (ω ) γγ 0 ) > d( x γ0, 0)/2. Ω with I (Q m, m 1) is a let Følner sequence in Γ then Q m Γ n γ 0 lim = Γ/Γ n 1. m Q m We conclude that the set {x (ω) : ω Ω} contains, or every m 1, a (Q m, d( x γ0, 0)/2)-separated set o cardinality 2 Qm Γnγ0, and hence that h(α ) Γ/Γ n 1 log 2 > 0. In order to ind out more about the actual value o h(α ) we take a look at the periodic points o α. Fix a subgroup Γ Γ and denote by (5.2) Fix Γ (X ) = {x X : α γ x = x or every γ Γ } the subgroup o Γ -invariant points in X. Clearly, Fix Γ (X ) is Γ -invariant, and Fix Γ (X ) is Γ-invariant i and only i Γ is a normal subgroup o Γ. Next

11 we set (5.3) EXPANSIVE ALGEBRAIC ACTIONS 11 L (Γ) Γ = {w L (Γ) : λ γ w = w or every γ Γ }, W Γ = W L (Γ) Γ, L (Γ, Z) Γ = L (Γ, Z) L (Γ) Γ. We write ξ : L (Γ, Z) X or the group homomorphism described in Proposition 4.2. Proposition 5.2. Let Γ be a countable residually inite group, ZΓ, and let α be the Γ-action on X deined in (3.5) (3.6). For every subgroup Γ Γ o inite index, (5.4) Fix Γ (X ) = ξ(l (Γ, Z) Γ ) = L (Γ, Z) Γ /ρ (L (Γ, Z) Γ ). Proo. From the equivariance o ξ it is clear that that ξ(l (Γ, Z) Γ ) Fix Γ (X ). Conversely, i x Fix Γ (X ), then we can ind w W Γ L (Γ) Γ with η(w) = x (c. (3.7)), and the point v = ρ w L (Γ, Z) Γ satisies that ξ(v) = x. This proves that ξ(l (Γ, Z) Γ ) = Fix Γ (X ), and Proposition 4.2 guarantees that ker(ξ) L (Γ, Z) Γ = ρ (L (Γ, Z) Γ ). It ollows that Fix Γ (X ) = ξ(l (Γ, Z) Γ ) = L (Γ, Z) Γ /ρ (L (Γ, Z) Γ ), as claimed. Corollary 5.3. I Γ \Γ < then Fix Γ (X ) = det(ρ L (Γ) Γ ). Proo. I the coset space Γ \Γ is inite then L (Γ) Γ = R Γ \Γ and we denote by ρ L (Γ) the restriction to Γ L (Γ) Γ = Z Γ \Γ o the linear map ρ : L (Γ) L (Γ). Then ρ (L (Γ, Z) Γ ) L (Γ, Z) Γ and the absolute value o the determinant det(ρ L (Γ) Γ ) is equal to L (X, Z) Γ /ρ (L (Γ, Z) Γ ) = X Γ. Remark 5.4. Since α is expansive, ker(ρ ) = 0 by Theorem 3.2, and the proo o Theorem 3.2 shows that there exists, or every pair o distinct points x, x X, a γ Γ with d(x γ, x γ) (3 1 ) 1 (c. (3.10)). I Γ Γ is a subgroup with inite index, and i Q Γ is a undamental domain o the right coset space Γ \Γ, i.e. a inite subset such that {γq: γ Γ } is a partition o Γ, then the preceding paragraph implies that Fix Γ (X ) is (Q, (3 1 ) 1 )-separated in the sense that there exists, or any two distinct points x, x X Γ, a γ Q with d(x γ, x γ) (3 1 ) 1. For the terminology used in our next proposition we reer to the beginning o this section and to Remark 5.4. Proposition 5.5 ([14]). Let Γ be a countable residually inite amenable group and let (Γ n, n 1) be a sequence o inite index normal subgroups with lim n Γ n = {1} (c. (5.1)). Then there exists, or every inite subset

12 12 CHRISTOPHER DENINGER AND KLAUS SCHMIDT K Γ and every ε > 0, an integer M = M(K, ε) 1 such that every Γ n with n M has a (K, ε)-invariant undamental domain Q n o the coset space Γ/Γ n. Proo. This is a slight reormulation o [14, Theorem 1] requiring only very minor changes in the proo. We describe these changes briely, using essentially the same notation and terminology as in [14]. Fix a inite set K Γ and ε > 0 and let ε and η be suiciently small positive numbers whose sizes will be clear by examining the course the proo. Choose N 1 with ( ) 1 ε N 2 < ε 200 and use the amenability o Γ to ind an increasing sequence (F j, j 1) such that (i) F 1 is (K, ε)-invariant and 1 F 1, (ii) or 1 j < N, F j+1 F j and F j+1 is (F j Fj 1, η)-invariant. Since n 1 Γ n = {1} we can ind an M 1 with Γ M F 1 N F N = {1}. We ix n M and write θ : Γ Γ/Γ n = G or the quotient map which is injective on F N. The argument in the proo o [14, Theorem 1] allows us to ind or appropriately chosen ε, η inite subsets {γ i,j : i = 1,..., m j, j = 1,..., N} and {γ i,j : i = 1,..., m j, j = 1,..., N} o Γ and, or each j = 1,..., N, sets F i,j F j, i = 1,..., m j, F i,j F j, i = 1,..., m j, such that the ollowing conditions are satisied. (1) For every i, j, F i,j (1 ε) F j and F i,j (1 ε) F j ; (2) For every i, j, the sets F i,j and F i,j are (K, ε/2)-invariant; (3) The sets θ(f i,j γ i,j ), i = 1,..., m j, j = 1,..., N, are disjoint; (4) The sets θ(γ i,j F i,j ), i = 1,..., m j, j = 1,..., N, are disjoint; (5) The sets E = satisy that N m j j=1 i=1 E (1 θ(f i,j γ i,j ), E = N m j j=1 i=1 θ(γ i,jf i,j) ε 100 ) G, E (1 ε 100 ) G. We set F = E E and choose a set Q F such that Q = G and θ(q) = G. Then Q is a (K, ε)-invariant undamental domain o the coset space Γ/Γ n. Corollary 5.6. Let Γ be a countable residually inite amenable group and let (Γ n, n 1) be a sequence o inite index normal subgroups with lim n Γ n =

13 EXPANSIVE ALGEBRAIC ACTIONS 13 {1}. Then there exists a Følner sequence (Q n, n 1) such that Q n is a undamental domain o Γ/Γ n or every n 1. Theorem 5.7. Let Γ be a countable residually inite amenable group and let (Γ n, n 1) be a sequence o inite index normal subgroups with lim n Γ n = {1}. I ZΓ, and i the algebraic Γ-action α on X in (3.5) (3.6) is expansive, then 1 h(α ) = lim n Γ/Γ n log Fix Γ n (X ) 1 (5.5) = lim n Γ/Γ n log L (Γ, Z) Γn /ρ (L (Γ, Z) Γn ) 1 = lim n Γ/Γ n log det(ρ L (Γ) Γn ), where Fix Γn (X ) X is the subgroup o Γ n -periodic points in (5.2) and L (Γ) Γn L (Γ) is deined in (5.3) (c. Corollary 5.3). Proo. We choose a Følner sequence (Q n, n 1) in Γ such that Q n is a undamental domain o Γ/Γ n or every n 1 (c. Corollary 5.6). Proposition 5.2 and Corollary 5.3 show that there exists, or every n 1, a (Q n, (3 1 ) 1 )-separated set (in the sense o Remark 5.4) o cardinality Fix Γn (X ) = L (Γ, Z) Γn /ρ (L (Γ, Z) Γn ) = det(ρ L (Γ) Γn ). Since (Q n, n 1) is Følner and Q n = Γ/Γ n this implies that h(α ) lim sup n 1 Q n log Fix Γ n (X ). Conversely, let δ > 0, ε < δ/3, and let F ε be a inite symmetric set with γ Γ F ε w γ < ε/ 1 (c. the proo o Theorem 4.4). The sets P n = Q n γ F ε Q n γ, n 1, orm a Følner sequence with lim n P n Q n = 1. We ix n 1 or the moment and choose a maximal set S n,δ X which is (P n, δ)-separated in the sense o Remark 5.4. For every x S n,δ we ind w(x) W L (Γ) with w(x) 1 /2 and η(w(x)) = x (Lemma 4.5) and write v(x) L (Γ, Z) Γn or the unique point with v(x) γ = (ρ w(x)) γ or every γ Q n. Our choice o F ε implies that the points {ξ(v(x)) : x S n,δ } Fix Γn (X ) are (P n, δ/3)-separated and thereore distinct, and Proposition 5.2 shows that S n,δ Fix Γn (X ). P Since (P n, n 1) is Følner and lim n n Q n = 1 this implies that h(α ) = lim n 1 P n log S n,δ lim in n which completes the proo o (5.5). 1 Q n log Fix Γ n (X ),

14 14 CHRISTOPHER DENINGER AND KLAUS SCHMIDT Corollary 5.8. Let Γ be a countable residually inite amenable group, and let, g ZΓ be elements such that the algebraic Γ-actions α and α g on X and X g in (3.5) (3.6) are expansive. I α and α g are given as in (3.5) (3.6) with replaced by and g, respectively, then α and α g are expansive, h(α ) = h(α ) and h(α g ) = h(α ) + h(α g ). Proo. The expansiveness o α and α g is clear rom Theorem 3.2, and the entropy ormulae ollow rom the usual properties o determinants (c. (5.5)). 6. Entropy and Fuglede Kadison determinants In this section L p (Γ, C) will denote the complex L p -space o Γ or 1 p with its conjugate linear involution w w given by (w ) γ = w γ 1, γ Γ. The von Neumann algebra N Γ o a discrete group Γ can be deined as the algebra o let Γ-equivariant bounded operators o L 2 (Γ, C) to itsel. Thus a bounded operator A on L 2 (Γ, C) belongs to N Γ i and only i we have or all v in L 2 (Γ, C) and all γ in Γ. λ γ A(v) = A(λ γ (v)) The homomorphism o C-algebras with involution: ρ : L 1 (Γ, C) N Γ mapping to the operator ρ with ρ (v) = v is injective because ρ (e(1)) =. In the ollowing we will sometimes view ρ as an inclusion both o L 1 (Γ, C) and o CΓ into N Γ and omit it rom the notation. The von Neumann trace on N Γ is the linear orm tr N Γ : N Γ C mapping A to tr N Γ (A) = A(e(1)), e(1). The trace is aithul in the sense that tr N Γ A = 0 or a positive operator A in N Γ implies that A = 0. Moreover tr N Γ vanishes on commutators and satisies the estimate tr N Γ A A. On L 1 (Γ, C) it is given by tr N Γ (w) = w(1). All this is easy to check. The Fuglede Kadison determinant o A in (N Γ) is the positive real number ( ) 1 (6.1) det N Γ A = exp 2 tr N Γ(log AA ). Note here that the operator AA is invertible and positive so that log AA is deined by the unctional calculus. I E λ is a spectral resolution or AA we can also write: (6.2) det N Γ A = exp ( ) log λ d tr N Γ (E λ ).

15 EXPANSIVE ALGEBRAIC ACTIONS 15 I the group Γ is inite we have N Γ = CΓ and (6.3) det N Γ A = deta 1/ Γ. We reer to [11] Example 3.13 or [3] Proposition 3.1 or a discussion o the case Γ = Z n, or see below. For positive A in (N Γ) we have det N Γ A = exp tr N Γ (log A). For operators 0 A B in (N Γ) this implies the inequality (6.4) det N Γ A det N Γ B. It is a non-trivial act that det N Γ deines a homomorphism det N Γ : (N Γ) R +. The intuitive reason or this is the Campbell Hausdor ormula. The actual proo in [7] or II 1 -actors and the analogous argument or group von Neumann algebras in [11, 3.2] is dierent though in order to avoid convergence problems. The ollowing remark on the disintegration o det N Γ serves only to give some urther background on this determinant. The required theory can be ound in [4] or example. Let Z(Γ) be the center o Γ. For a character χ in Ẑ(Γ) consider the Hilbert space L 2 (Γ, C) χ = {h in L 2 (Γ, C) : ρ γ h = χ(γ)h or all γ in Z(Γ)}. By the spectral theorem applied to the commuting unitary operators ρ γ with γ in Z(Γ) we have a direct integral decomposition L 2 (Γ, C) = L 2 (Γ, C) χ dµ(χ) Ẑ(Γ) where µ is the Haar probability measure on the compact abelian group Ẑ(Γ). Correspondingly we get a disintegration N Γ = N χ Γ dµ(χ) Ẑ(Γ) where N χ Γ is the von Neumann algebra o bounded operators on L 2 (Γ, C) χ which commute with the let Γ-action λ. The Fuglede Kadison determinant det NχΓ is deined similarly as above and we get the ormula log det N Γ A = log det NχΓA χ dµ(χ). Ẑ(Γ)

16 16 CHRISTOPHER DENINGER AND KLAUS SCHMIDT Here A χ is the restriction o A to an invertible operator rom L 2 (Γ, C) χ to itsel. I Γ is commutative we ind that log det N Γ A = log a(χ) dµ(χ), ˆΓ i A χ acts on the 1-dimensional space L 2 (Γ, C) χ by multiplication with a(χ) C. For in CΓ in particular we get (6.5) log det N Γ = log ˆ(χ) dµ(χ) ˆΓ where ˆ is the Fourier transorm o. We now begin with some preparations or the proo o the entropy ormula in Theorem 1.1. Let Γ be a countable residually inite discrete group and let (Γ n, n 1) be a sequence o inite index normal subgroups with lim n Γ n = {1}. Set Γ (n) = Γ n /Γ. For in L 1 (Γ, C) the bounded operator ρ : L 2 (Γ, C) L 2 (Γ, C) given by right convolution with satisies the norm estimate (6.6) ρ 1. The group Γ acts via λ on L (Γ, C) and we have an isomorphism o inite dimensional C-vector spaces (6.7) L (Γ, C) Γn = L (Γ (n), C) given by viewing let Γ n -invariant unctions on Γ as unctions on Γ n /Γ. Since ρ is let Γ n -equivariant it induces an endomorphism o L (Γ, C) Γn and hence an endomorphism o L (Γ (n), C) = CΓ (n) = L 2 (Γ (n), C) which we denote by: (6.8) ρ (n) : L 2 (Γ (n), C) L 2 (Γ (n), C). Consider the map integration along the ibres : (6.9) L 1 (Γ, C) L 1 (Γ (n), C) given by sending : Γ C to the unction (n) : Γ (n) C deined by (n) (δ) = γ δ (γ), or all residue classes in δ in Γ (n). As short calculation shows that (6.9) is a homomorphism o C-algebras with involution such that (n) 1 1. Moreover we have (6.10) ρ (n) = ρ (n) : L 2 (Γ (n), C) L 2 (Γ (n), C). By the estimate (6.6) applied to (n) and Γ (n) we have ρ (n) (n) 1. Using (6.10) we get the uniorm estimate or all n 1 (6.11) ρ (n) 1.

17 From the deinition o ρ (n) EXPANSIVE ALGEBRAIC ACTIONS 17 the relation ρ (n) g invertible i is invertible in L 1 (Γ, C) and we have (ρ (n) with (6.11) this gives the estimate = ρ(n) ρ (n) g ollows. Hence ρ (n) is ) 1 = ρ (n). Together 1 (6.12) (ρ (n) ) or in L 1 (Γ, C) and all n 1. Note that by equation (6.3) we have (6.13) log det N Γ (n) (n) 1 = Γ/Γ n log det ρ (n) = 1 Γ/Γ n log det(ρ L (Γ,C) Γn ). Hence Theorem 1.1 o the introduction ollows rom Theorem 3.2, Theorem 5.7 and the ollowing result: Theorem 6.1. Let Γ be a countable residually inite discrete group and (Γ n, n 1) a sequence o inite index normal subgroups with lim n Γ n = {1}. For in L 1 (Γ, C) we have det N Γ = lim n det N Γ (n) (n). Proo. Because o the relation ( ) (n) = (n) (n) the assertion means: (6.14) tr N Γ log ρ g = lim n tr N Γ (n) log ρ g (n) or g = in L 1 (Γ, C). But ρ g = ρ ρ and ρ g (n) = ρ (n)ρ are positive (n) operators on L 2 (Γ, C). By the estimate (6.6) applied to g and g 1 and the estimates (6.11) and (6.12) applied to g instead o it ollows that the spectra σ(ρ g ) and σ(ρ g (n)) lie in the closed interval I = [ g 1 1, g 1]. Fix ε > 0. By the Weierstrass approximation theorem there exists a real polynomial Q such that sup log t Q(t) ε. t I Since the spectra o ρ g and ρ g (n) lie in I it ollows that we have log ρ g Q(ρ g ) ε and log ρ g (n) Q(ρ g (n)) ε. Using the estimate tr N Γ A A we ind: tr N Γ log ρ g tr N Γ (n) log ρ g (n) tr N Γ (log ρ g Q(ρ g )) + tr N Γ Q(ρ g ) tr N Γ (n)q(ρ g (n)) + tr N Γ (n)(log ρ g (n) Q(ρ g (n))) log ρ g Q(ρ g ) + tr N Γ Q(ρ g ) tr N Γ (n)q(ρ g (n)) + log ρ g (n) Q(ρ g (n)) 2ε + tr N Γ Q(g) tr N Γ (n)q(g (n) ). Hence ormula (6.14) is a consequence o the ollowing Lemma 6.2. Lemma 6.2. For any in L 1 (Γ, C) and any complex polynomial Q(t) we have tr N Γ Q() = lim n tr N Γ (n)q( (n) ).

18 18 CHRISTOPHER DENINGER AND KLAUS SCHMIDT Proo. Since Q() is in L 1 (Γ, C) and Q() (n) = Q( (n) ) it suices to prove the assertion or Q(t) = t i.e. that tr N Γ = lim tr (n) n N Γ (n) or all in L 1 (Γ, C). Writing = γ γe(γ), we have (n) = γ γe(γ) where γ = Γ n γ. Thus we get tr N Γ = 1 and tr N Γ (n) (n) = γ Γ n γ. Fix some ε > 0. Since is in L 1 (Γ, C) we have γ Γ γ <. Hence there is a inite subset K o Γ with 1 K such that we have γ Γ K γ < ε. Since lim n Γ n = {1} we can ind an index N 1 such that Γ n K 1 K = {1} or all n N. Since 1 K it ollows that Γ n K = {1} or all n N as well. For n N we thereore get the estimate: tr N Γ tr N Γ (n) (n) = 1 γ γ γ Γ n γ Γ K Since ε > 0 was arbitrary the lemma is proved. γ < ε. γ Γ n {1} Remark 6.3. For in ZΓ the element (n) lies in ZΓ (n). Hence tr N Γ and tr N Γ (n) (n) are integers and it ollows that we have tr N Γ = tr N Γ (n) (n) or all n 0, c.. [1] Lemma 2.2 or [12] 5.5. Lemma. For an element in L 1 (Γ, C) let σ = lim n n 1/n 1 1 be the spectral radius o. We have Hence log det N Γ = 1 2 tr N Γ log ρ 1 2 log ρ (6.6) 1 2 log 1 log 1. (6.15) log det N Γ = 1 n log det N Γ n log n 1/n 1. Thereore log det N Γ log σ. I is in L 1 (Γ, C) inequality (6.15) applied to 1 gives: and in the limit log det N Γ log n 1/n 1 log det N Γ log σ 1. Together with Theorem 1.1 this gives the ollowing corollary: Corollary 6.4. In the situation o Theorem 1.1, i ZΓ is a unit in L 1 (Γ, C) we have the estimates log log(σ 1) 1 h log σ log 1.

19 EXPANSIVE ALGEBRAIC ACTIONS 19 Example 6.5. For Γ = Z n the spectrum o the commutative normed - algebra L 1 (Z n ) is the real n-torus T n = S n, where S = {z C : z = 1} (c.. [15, XI 2. Example 2]). Hence an element in L 1 (Z n ) is invertible i and only i the Fourier series ˆ does not vanish in any point o S n (Wiener s theorem). By [15] XI 2. Theorem 1, we have σ = max x S n ˆ(x) and σ 1 = max x S n 1 (x) = max x S n ˆ(x) 1. For an element in Z[Z n ] which is invertible in L 1 (Z n ) the corollary thereore gives the estimates: min log ˆ(x) = log(σ 1) 1 h log σ = max log ˆ(x). x S n n O course, this estimate ollows directly rom the ormula [10] h = log ˆ(x) dµ(x), S n which in our case is also a special case o Theorem 1.1 and equation (6.5). Corollary 6.6. In the situation o Theorem 1.1, consider elements, g in ZΓ L 1 (Γ, C) which satisy 0 g in N Γ. Then we have h h g and equality h = h g holds i and only i = g. Proo. The irst assertion ollows rom inequality (6.4) applied to the operators ρ and ρ g on L 2 (Γ, C). By the monotonicity o the logarithm the operator A = log ρ g log ρ in N Γ is positive. I the entropies are equal, i.e. i h = h g, then it ollows rom Theorem 1.1 that tr N Γ (log ρ ) = tr N Γ (log ρ g ). But this means that tr N Γ A = 0 and hence A = 0 since tr N Γ is aithul. Applying exp to the equality log ρ = log ρ g gives the second assertion. The ollowing is an application o our theory to the structure o (N Γ). Corollary 6.7. For a group Γ as in Theorem 1.1 let be an element o ZΓ which is invertible in L 1 (Γ) but does not have a let inverse in ZΓ. Then is not contained in the commutator subgroup o (N Γ). Proo. Otherwise we would have det N Γ = 1 and hence h = 0 by Theorem 1.1. By assumption we have ZΓ ZΓ and hence X 0. But then Theorem 5.1 tells us that h > 0, contradiction. x S Reerences [1] B. Clair, Residual amenability and the approximation o L 2 -invariants, Michigan Math. J. 46 (1999), [2] Y. Choi, Injective convolution operators on L (Γ) are surjective, arxiv:math.fa/ v1 15 Jun [3] C. Deninger, Fuglede Kadison determinants and entropy or actions o discrete amenable groups, J. Amer. Math. Soc. 19 (2006), [4] J. Dixmier, Von Neumann algebras, North Holland, Amsterdam, [5] N. Dunord and J.T. Schwartz, Linear operators, vol. 1, Wiley, New York, 1967.

20 20 CHRISTOPHER DENINGER AND KLAUS SCHMIDT [6] M. Einsiedler and H. Rindler, Algebraic actions o the discrete Heisenberg group and other non-abelian groups, Aequationes Math. 62 (2001), [7] B. Fuglede and R.V. Kadison, Determinant theory in inite actors, Ann. Math. 55 (1952), [8] I. Kaplansky, Fields and rings, University o Chicago Press, Chicago-London, [9] D. Lind and K. Schmidt, Homoclinic points o algebraic Z d -actions, J. Amer. Math. Soc. 12 (1999), [10] D. Lind, K. Schmidt and T. Ward, Mahler measure and entropy or commuting automorphisms o compact groups, Invent. Math. 101 (1990), [11] W. Lück, L 2 -invariants: Theory and applications to geometry and K-theory, Springer, Berlin, [12] T. Schick, L 2 -determinant class and approximation o L 2 -Betti numbers, TAMS 353 (2001), [13] K. Schmidt, Dynamical systems o algebraic origin, Birkhäuser, Basel [14] B. Weiss, Monotileable amenable groups, Amer. Math. Soc. Transl. 202 (2001), [15] K. Yosida, Functional analysis, Springer, Heidelberg, Christopher Deninger: Mathematical Institute, West. Wilhelms-University Münster, Einsteinstr. 62, Münster, Germany address: c.deninger@math.uni-muenster.de Klaus Schmidt: Mathematics Institute, University o Vienna, Nordbergstraße 15, A-1090 Vienna, Austria and Erwin Schrödinger Institute or Mathematical Physics, Boltzmanngasse 9, A-1090 Vienna, Austria address: klaus.schmidt@univie.ac.at