p-adic entropy and a p-adic Fuglede–Kadison determinant

Christopher Deninger

Mathematisches Institut, Einsteinstr. 62, 48149 Münster, Germany

c.deninger@math.uni-muenster.de

Dedicated to Yuri Ivanovich Manin

1 Introduction

In several instances the entropy h(φ) of an automorphism φ on a space X can be calculated in terms of periodic points:

h(φ) = lim_{n→∞} \frac{1}{n} log |Fix (φ^{n})| .  (1)

Here Fix (φ^{n}) is the set of fixed points of φ^{n} on X. Let log_{p} : ℚ_{p}^{∗} → ℤ_{p} be the branch of the p-adic logarithm normalized by log_{p}(p) = 0. The p-adic analogue of the limit (1), if it exists, may be viewed as a kind of entropy with values in the p-adic number field ℚ_{p},

h_{p}(φ) = lim_{n→∞} \frac{1}{n} log_{p} |Fix (φ^{n})| .  (2)

It depends only on the action of φ on X viewed as a set.

An earlier different approach to a p-adic entropy theory was mentioned to me by Amnon Besser. The usual definitions of measure theoretic or topological entropy have no obvious p-adic analogue since \overline{lim} or sup do not make sense p-adically and since the cardinalities of partitions, coverings and of separating or spanning sets do not behave reasonably in the p-adic metric.

Instead of actions of a single automorphism φ we look more generally at actions of a countable discrete residually finite but not necessarily amenable group Γ on a set X. Let us write Γ_{n} → e if (Γ_{n}) is a sequence of cofinite normal subgroups of Γ such that only the neutral element e of Γ is contained in infinitely many Γ_{n}’s. Let Fix_{Γ_{n}}(X) be the set of points in X which are fixed by Γ_{n}. If the limit:

h_{p} := lim_{n→∞} \frac{1}{(Γ : Γ_{n})} log_{p} |Fix_{Γ_{n}}(X)|  (3)

exists with respect to a choice of Γ_{n} → e we call it the p-adic entropy of the Γ -action on the set X (with respect to the sequence (Γ_{n})).

In this note we show that for an interesting class of Γ -actions the p-adic entropy exists independently of the choice of Γ_{n} → e. In these examples X is an abelian group and Γ acts by automorphisms of groups. Namely, let (ℝ/ℤ)^{Γ} be the full shift on Γ with values in the circle ℝ/ℤ and left Γ -action by γ(x_{γ′}) = (x_{γ^{−1}γ′}). For an element f = ∑_{γ} a_{γ}γ in the integral group ring ℤΓ consider the closed subshift X_{f} ⊂ (ℝ/ℤ)^{Γ} consisting of all sequences (x_{γ′}) which satisfy the equation

∑_{γ′} x_{γ′}a_{γ^{−1}γ′} = 0 in (ℝ/ℤ)^{Γ} for all γ ∈ Γ .

In fact as in [ER01] we study more general systems defined by an r ×r-matrix over ℤΓ. However, in this introduction, for simplicity, we describe only the case r = 1. If Γ is amenable, we denote by h(f) the topological entropy of the Γ -action on X_{f}.

The case Γ = ℤ^{d} is classical. Here we may view f as a Laurent polynomial and according to [LSW90] the entropy is given by the (logarithmic) Mahler measure of f

h(f) = m(f) := ∫_{T^{d}} log |f(z)| dμ(z) .  (4)

Here μ is the normalized Haar measure on the d-torus T^{d}. According to [LSW90] the ℤ^{d}-action on X_{f} is expansive if and only if f does not vanish in any point of T^{d}. By a theorem of Wiener this is also equivalent to f being a unit in L^{1}(ℤ^{n}). In this case h(f) can be calculated in terms of periodic points, c.f. [LSW90] Theorem 7.1. See also [Sch95] for this theory.

What about a p-adic analogue? In [Den97] it was observed that in the expansive case m(f) has an interpretation via the Deligne–Beilinson regulator map from algebraic K-theory to Deligne cohomology. Looking at the analogous regulator map from algebraic K-theory to syntomic cohomology one gets a suggestion what a (purely) p-adic Mahler measure m_{p}(f) of f should be, c.f. [BD99]. It can only be defined if f does not vanish in any point of the p-adic d-torus T_{p}^{d} = {z ∈ ℂ_{p}^{d} | |z_{i}|_{p} = 1} , where ℂ_{p} is the completion of a fixed algebraic closure \overline{ℚ}_{p} of ℚ_{p}. In this case m_{p}(f) is given by the convergent Snirelman integral

m_{p}(f) = ∫_{T_{p}^{d}} log_{p} f(z) .  (5)

Recall that the Snirelman integral of a continuous function F : T_{p}^{d} → ℂ_{p} is defined by the following limit if it exists:

∫_{T_{p}^{d}} F (z) := lim_{{N→∞ \atop (N,p)=1}} \frac{1}{N^{d}} ∑_{ζ∈μ_{N}^{d}} F (ζ) .

Here μ_{N} is the group of N-th roots of unity in \overline{ℚ}_{p}^{∗}.

For example, let P (t) = a_{m}t^{m} + … + a_{r}t^{r} be a polynomial in ℂ_{p}[t] with a_{m}, a_{r} ≠ 0 whose zeroes α satisfy |α|_{p} ≠ 1. Then, according to [BD99] Proposition 1.5 we have the following expression for the p-adic Mahler measure:

m_{p}(f) = log_{p} a_{r} − ∑_{0<|α|_{p}<1} log_{p} α  (6)
= log_{p} a_{m} + ∑_{|α|_{p}>1} log_{p} α .

For d ≥ 2 there does not seem to be a simple formula for m_{p}(f).

In [BD99] we mentioned the obvious problem to give an interpretation of m_{p}(f) as a p-adically valued entropy. This is now provided by the following result:

Theorem 1. Assume that f ∈ ℤ[ℤ^{d}] = ℤ[t_{1}^{±1}, …, t_{d}^{±1}] does not vanish in any point of the p-adic d-torus T_{p}^{d}. Then the p-adic entropy h_{p}(f) of the Γ = ℤ^{d}-action on X_{f} in the sense of (3) exists for all Γ_{n} → 0 and we have h_{p}(f) = m_{p}(f).

Now we turn to more general groups Γ. In [DS] extending [Den06] it was shown that for countable residually finite amenable groups Γ and elements f in ℤΓ which are invertible in L^{1}(Γ) we have

h(f) = log det_{N Γ} f .  (7)

Here det_{N Γ} is the Fuglede–Kadison determinant [FK52] on the units of the von Neumann algebra N Γ ⊃ L^{1}Γ ⊃ ℤΓ of Γ. In fact, equation (7) holds without the condition of amenability if h(f) is replaced by the quantity:

h_{per} (f) := lim_{n→∞} \frac{1}{(Γ : Γ_{n})} log |Fix_{Γ_{n}}(X)| .

For the Γ -action on X_{f} this limit exists and is independent of the choice of sequence Γ_{n} → e.

In the p-adic case, instead of working with a p-adic L^{1}-convolution algebra it is more natural to work with the bigger convolution algebra c_{0}(Γ). It consists of all formal series x = ∑_{γ} x_{γ}γ with x_{γ} ∈ ℚ_{p} and |x_{γ}|_{p} → 0 as γ → ∞ in Γ.

For Γ = ℤ^{d} it is known that f ∈ ℤ[ℤ^{d}] does not vanish in any point of the p-adic d-torus T_{p}^{d} if and only if f is a unit in the algebra c_{0}(ℤ^{d}). Hence in general, it is natural to look for a p-adic analogue of formula (7) for all f ∈ ℤΓ which are units in c_{0}(Γ). In the p-adic case there is no analogue for the theory of von Neumann algebras and for the functional calculus used to define det_{N Γ}. However using some algebraic K-theory and the results of [FL03], [BLR] and [KLM88] we can define a p-adic analogue log_{p} det_{Γ} of log det_{N Γ} for suitable classes of groups Γ. For example we get the following result generalizing theorem 1:

Theorem 2. Assume that the residually finite group Γ is elementary amenable and torsion-free. Let f be an element of ℤΓ which is a unit in c_{0}(Γ). Then the p-adic entropy h_{p}(f) of the Γ -action on X_{f} in the sense of (3) exists for all Γ_{n} → e and we have

h_{p}(f) = log_{p} det_{Γ} f .

Acknowledgement: I would like to thank my colleagues Wolfgang Lück and Peter Schneider for helpful conversations.

2 Preliminaries

Fix an integer r ≥ 1 and set T = T^{r} = (ℝ/ℤ)^{r}. For a discrete group Γ let T^{Γ} be the full shift with left Γ -action by γ(x_{γ′}) = (x_{γ^{−1}γ′}). Write M_{r}(R) for the ring of r × r-matrices over a ring R. For an element f = ∑ a_{γ}γ in M_{r}(ℤ)[Γ] = M_{r}(ℤΓ) the closed subshift X_{f} ⊂ T^{Γ} is defined as the closed subgroup consisting of all sequences with

∑_{γ′} x_{γ′}a_{γ^{−1}γ′}^{∗} = 0 in T^{Γ} for all γ ∈ Γ .

Here a^{∗} denotes the transpose of a matrix a in M_{r}(ℤ). The group ring M_{r}(ℤ)[Γ] is equipped with an anti-involution ∗ defined by f^{∗} = ∑_{γ} a_{γ^{−1}}^{∗}γ for f = ∑_{γ} a_{γ}γ.

Let ρ_{f} be right multiplication by f^{∗} on the group T [[Γ]] of formal T -valued series on Γ. For x = ∑_{γ} x_{γ}γ in T [[Γ]] we have

ρ_{f}(x) = ∑_{γ} x_{γ}γ ∑_{γ} a_{γ^{−1}}^{∗}γ = ∑_{γ} ( ∑_{γ′} x_{γ′}a_{γ^{−1}γ′}^{∗})γ .

Hence we see that

X_{f} = Ker (ρ_{f} : T [[Γ]] ⟶ T [[Γ]])

where on the right hand side the group Γ acts by left multiplication. Let N be a normal subgroup of Γ with quotient map ∼: Γ → Γ̃ = Γ/N. Set

f̃ = ∑_{γ} a_{γ} γ̃ = ∑_{δ∈Γ̃} ( ∑_{γ∈δ} a_{γ})δ in M_{r}(ℤ)[Γ̃] .

This is the image of f under the reduction map M_{r}(ℤ)[Γ] → M_{r}(ℤ)[Γ̃]. Under the natural isomorphism

T [[Γ̃]] ⟶^{∼} Fix_{N}(T [[Γ]])

mapping ∑_{δ} x_{δ}δ to ∑_{γ} xγ̃γ the action ρf̃ corresponds to the restriction of ρ_{f}. Hence we have

Fix_{N}(X_{f}) = Ker (ρf̃: T [[Γ̃]] ⟶ T [[Γ̃]]) = Xf̃ .

If we assume that Γ̃ is finite we get that

Fix_{N}(X_{f}) = ρ_{f̃,ℝ}^{−1}(ℤΓ̃)^{r}/(ℤΓ̃)^{r}

for the endomorphism ρ_{f̃,ℝ} of right multiplication by f̃^{∗} on (ℝΓ̃)^{r}. This implies the following fact, c.f. [DS], Corollary 4.3:

Proposition 3. Let Γ̃ be finite. Then ρf̃ is an isomorphism of (ℚΓ̃)^{r} if and only if Fix_{N}(X_{f}) is finite. In this case the order is given by

|Fix_{N}(X_{f})| = ±detρf̃ .

This follows from the fact that for an isomorphism φ of a finite dimensional real vector space V and a lattice Λ in V with φ(Λ) ⊂ Λ we have:

|φ^{−1}Λ/Λ| = |Λ/φ(Λ)| = | det(φ | V)| .

For any countable discrete group Γ let c_{0}(Γ) be the set of formal series ∑_{γ} x_{γ}γ with x_{γ} ∈ ℚ_{p} and |x_{γ}|_{p} → 0 for γ → ∞. This means that for any ε > 0 there is a finite subset S ⊂ Γ such that |x_{γ}|_{p} < ε for all γ ∈ Γ \ S. The set c_{0}(Γ) is a ℚ_{p}-vector space and it becomes a ℚ_{p}-algebra with the product

∑_{γ} x_{γ}γ · ∑_{γ} y_{γ}γ = ∑_{γ} ( ∑_{γ′γ′′=γ} x_{γ′}y_{γ′′})γ .  (8)

Note that the sums

∑_{γ′γ′′=γ} x_{γ′}y_{γ′′} = ∑_{γ′} x_{γ′}y_{γ^{′−1}γ}

converge p-adically for every γ since lim_{γ′→∞} |x_{γ′}y_{γ^{′−1}γ}|_{p} = 0. The value is independent of the order of summation. Moreover, because of the inequality

| ∑_{γ′} x_{γ′}y_{γ^{′−1}γ} |_{p} ≤ sup_{γ′} |x_{γ′}y_{γ^{′−1}γ}|_{p} ,  (9)

we have

lim_{γ→∞} ∑_{γ′} x_{γ′}y_{γ^{′−1}γ} = 0 ,

so that the product (8) is well defined. We may also view c_{0}(Γ) as an algebra of ℚ_{p}-valued functions on Γ under convolution.

The ℚ_{p}-algebra c_{0}(Γ) is complete in the norm

∥ ∑_{γ} x_{γ}γ∥ = sup_{γ} |x_{γ}|_{p} = max_{γ} |x_{γ}|_{p} .

The norm satisfies the following properties:

∥x∥ = 0 if and only if x = 0  (10)
∥x + y∥ ≤ max(∥x∥, ∥y∥)  (11)
∥λx∥ = |λ|_{p} ∥x∥ for all λ ∈ ℚ_{p}  (12)
∥xy∥ ≤ ∥x∥ ∥y∥ and ∥1∥ = 1  (13)

Hence c_{0}(Γ) is a p-adic Banach algebra over ℚ_{p}, i.e. a unital ℚ_{p}-algebra B which is complete with respect to a norm ∥ ∥ : B → ℝ^{≥0} satisfying conditions (10)–(13).

We will only consider Banach algebras where ∥ ∥ takes values in p^{ℤ} ∪ {0}. The subring A = B^{0} of elements x in B of norm ∥x∥ ≤ 1 is a p-adic Banach algebra over ℤ_{p}, defined similarly as before. An example is given by

c_{0}(Γ, ℤ_{p}) = c_{0}(Γ)^{0} = {∑ x_{γ}γ | x_{γ} ∈ ℤ_{p} with lim_{γ→∞} |x_{γ}|_{p} = 0} .

In this case the residue algebra A/pA over 𝔽_{p} is isomorphic to the group ring of Γ over 𝔽_{p}:

c_{0}(Γ, ℤ_{p})/pc_{0}(Γ, ℤ_{p}) = 𝔽_{p}[Γ] .  (14)

The 1-units U^{1} = 1 + pA form a subgroup of A^{∗} since

(1 + pa)^{−1} := ∑_{ν=0}^{∞}(−pa)^{ν}

provides an inverse of 1 + pa ∈ U^{1} in U^{1}. It is easy to see that one has an exact sequence of groups

1 ⟶ U^{1} ⟶ A^{∗} ⟶ (A/pA)^{∗} ⟶ 1 .  (15)

For A = c_{0}(Γ, ℤ_{p}) this is the exact seqence

1 ⟶ 1 + pc_{0}(Γ, ℤ_{p}) ⟶ c_{0}(Γ, ℤ_{p})^{∗} ⟶ 𝔽_{p}[Γ]^{∗} ⟶ 1 .  (16)

Concerning the units of a p-adic Banach algebra over ℚ_{p}, we have the following known fact:

Proposition 4. Let B be a p-adic Banach algebra over ℚ_{p} whose norm takes values in p^{ℤ} ∪ {0} and set A = B^{0}. If the residue algebra A/pA has no zero divisors, then we have

B^{∗} = p^{ℤ}A^{∗} and p^{ℤ} ∩ A^{∗} = 1 .

Proof. For f in B^{∗} set g = 1/f. Let ν, μ be such that f_{1} = p^{ν}f and g_{1} = p^{μ}g have norm one. The reductions \overline{f}_{1}, \overline{g}_{1} of f_{1}, g_{1} are non-zero. In the equation f_{1}g_{1} = p^{ν+μ} we have ν + μ ≥ 0. Reducing mod p we find that 0 ≠ \overline{f}_{1}\overline{g}_{1} = p^{ν+μ} mod p in A/pA. Hence we have ν+μ = 0 and therefore f_{1}g_{1} = 1. The first assertion follows. Because of (13) we have ∥a∥ = 1 for a ∈ A^{∗} and ∥p^{ν}∥ = p^{−ν}. This implies the second assertion. ⊓ ⊔

Example 5. For the group Γ = ℤ^{d} the algebra c_{0}(ℤ^{d}) can be identified with the affinoid commutative algebra ℚ_{p}⟨t_{1}^{±1}, …, t_{d}^{±1}⟩ of power series ∑_{ν∈ℤ^{d}} x_{ν}t^{ν} with x_{ν} ∈ ℚ_{p} and lim_{|ν|→∞} |x_{ν}|_{p} = 0. Note that these power series can be viewed as functions on T_{p}^{d}. The residue algebra is 𝔽_{p}[ℤ^{d}] = 𝔽_{p}[t_{1}^{±1}, …, t_{d}^{±1}]. It has no zero divisors and its groups of units is

𝔽_{p}[ℤ^{d}]^{∗} = 𝔽_{p}^{∗}t_{1}^{ℤ} ⋯ t_{d}^{ℤ} .

The preceeding proposition and the exact sequence (16) now give a decomposition into a direct product of groups

c_{0}(ℤ^{d})^{∗} = p^{ℤ}μ_{p−1}t_{1}^{ℤ} ⋯ t_{d}^{ℤ}(1 + p c_{0}(ℤ^{d}, ℤ_{p})) .

Proposition 6. For f in ℚ_{p}[ℤ^{d}] = ℚ_{p}[t_{1}^{±1}, …, t_{d}^{±1}] the following properties are equivalent:

a) We have f(z) ≠ 0 for every z in T_{p}^{d}

b) f is a unit in c_{0}(ℤ^{d})^{∗}

c) f has the form f(t) = ct^{ν}(1 + pg(t)) for some c ∈ ℚ_{p}^{∗}, ν ∈ ℤ^{d} and g(t) in c_{0}(ℤ^{d}, ℤ_{p}).

Proof. We have seen that b) and c) are equivalent and it is clear that both b) and c) imply a). For proving that a) implies b) note that the maximal ideals of c_{0}(ℤ^{d}) = ℚ_{p}⟨t_{1}^{±1}, …, t_{d}^{±1}⟩ correspond to the orbits of the Gal (\overline{ℚ}_{p}/ℚ_{p})- operation on T_{p}^{d} ∩ (\overline{ℚ}_{p}^{∗})^{d}. Hence f is not contained in any maximal ideal of c_{0}(ℤ^{d}) by assumption a) and therefore f is a unit. ⊓ ⊔

3 The Frobenius group determinant and a proof of Theorem 1

The map ℤΓ → ℤΓ̃ for Γ̃ = Γ/N from the beginning of the last section can be extended to a homomorphism of ℚ_{p}-algebras c_{0}(Γ) → c_{0}(Γ̃) by sending f = ∑ a_{γ}γ to f̃ = ∑ a_{γ} γ̃. Note that this is well defined by the ultrametric inequality and that we have ∥f̃∥ ≤ ∥f∥. If Γ̃ is finite, we have c_{0}(Γ̃) = ℚ_{p}Γ̃ and hence we obtain a homomorphism of groups GL_{r}(c_{0}(Γ)) → GL_{r}(ℚ_{p}Γ̃). It follows that for f in M_{r}(ℤΓ) ∩ GL_{r}(c_{0}(Γ)) the endomorphism ρf̃ of (ℚΓ̃)^{r} is an isomorphism. Together with proposition 3 we have shown the first equation in the following proposition:

Proposition 7. Let Γ be a discrete group and N a normal subgroup with finite quotient group Γ̃. For f in M_{r}(ℤΓ) ∩ GL_{r}(c_{0}(Γ)) the set Fix_{N}(X_{f}) is finite and we have

|Fix_{N}(X_{f})| = ± det ρf̃
= ± ∏_{π} det_{\overline{ℚ}_{p}}( ∑_{γ} a_{γ}^{∗} ⊗ ρ_{π}(γ̃))^{d_{π}} .

Here π runs over the equivalence classes of irreducible representations ρ_{π} of Γ̃ on \overline{ℚ}_{p}-vector spaces V_{π} and d_{π} is the degree dim V_{π} of π.

Proof. It remains to prove the second equation which is essentially due to Frobenius. Consider \overline{ℚ}_{p}Γ̃ as a representation of Γ̃ via the map γ̃ ↦ ρ(γ̃) := right multiplication with γ̃^{−1}. This (“right regular”) representation decomposes as follows into irreducible representations c.f. [Sch95] I, 2.2.4

\overline{ℚ}_{p}Γ̃ ≅ ⨁_{π} V_{π}^{d_{π}} .

The endomorphism

ρf̃ = ∑_{γ} a_{γ}^{∗} ⊗ ρ(γ̃)

on

(\overline{ℚ}_{p}Γ̃)^{r} = \overline{ℚ}_{p}^{r} ⊗ \overline{ℚ}_{p}Γ̃

therefore corresponds to the endomorphism

⨁_{π} ( ∑_{γ} a_{γ}^{∗} ⊗ ρ_{π}(γ̃))^{d_{π}} on ⨁_{π} \overline{ℚ}_{p}^{r} ⊗ V_{π}^{d_{π}} = ⨁_{π}(\overline{ℚ}_{p}^{r} ⊗ V_{π})^{d_{π}} .

Hence the formula follows. ⊓ ⊔

Remark 8. In the real case and for the Heisenberg group, Klaus Schmidt previously used the group determinant to calculate |Fix_{Γ_{n}}(X_{f})| for f in L^{1}(Γ)^{∗} and certain Γ_{n}.

The following result generalizes Theorem 1 from the introduction, at least for a particular sequence Γ_{n} → 0:

Theorem 9. Let f = ∑_{ν∈ℤ^{n}} a_{ν}t^{ν} in M_{r}(ℤ[t_{1}^{±1}, …, t_{d}^{±1}]) be invertible in every point of the p-adic d-torus T_{p}^{d}. Then the p-adic entropy h_{p}(f) of the Γ = ℤ^{d}-action on X_{f} exists in the sense of (3) for the sequence Γ_{n} = (nℤ)^{d} → 0 with n prime to p, and we have

h_{p}(f) = m_{p}(det f) .

Proof. By assumption, the Laurent polynomial det f does not vanish in any point of T_{p}^{d}. Hence det f is a unit in c_{0}(Γ) by proposition 6. It follows from Proposition 7 that we have:

|Fix_{Γ_{n}}(X_{f})| = ± ∏_{χ} det_{\overline{ℚ}_{p}}( ∑_{ν∈ℤ^{d}} a_{ν}^{∗} ⊗ χ(ν))

where χ runs over the characters of Γ/Γ_{n} = (ℤ/nℤ)^{d}. These correspond via χ(ν) = ζ^{ν} to the elements ζ of μ_{n}^{d}. Viewing f as a matrix of functions on T_{p}^{d} we therefore get the formulas

|Fix_{Γ_{n}}(X_{f})| = ± ∏_{ζ∈μ_{n}^{d}} det_{\overline{ℚ}_{p}}( ∑_{ν∈ℤ^{d}} a_{ν}^{∗}ζ^{ν})

= ± ∏_{ζ∈μ_{n}^{d}} det_{\overline{ℚ}_{p}}(f(ζ))

= ± ∏_{ζ∈μ_{n}^{d}}(det f)(ζ) .

Thus the p-adic entropy h_{p}(f) of the Γ -action on X_{f} with respect to the above sequence is given by:

h_{p}(f) = lim_{{n→∞ \atop (n,p)=1}} \frac{1}{(Γ : Γ_{n})} log_{p} |Fix_{Γ_{n}}(X_{f})|
= lim_{{n→∞ \atop (n,p)=1}} \frac{1}{n^{d}} ∑_{ζ∈μ_{n}^{d}} log_{p}(det f)(ζ)

= ∫_{T_{p}^{d}} log_{p} det f = m_{p}(det f) .

Note here that for the Laurent polynomials det f under consideration the Snirelman integral exists by [BD99] Proposition 1.3. ⊓ ⊔

Remark 10. A suitable generalization of that proposition would give theorems 9 and 1 for general sequences Γ_{n} → 0 in Γ = ℤ^{d}. We leave this to the interested reader since the general case of Theorem 1 is also a corollary of Theorem 2 which will be proved by a different method in Section 5.

Example 11. The polynomial in one variable f(T) = 2T^{2} − T + 2 does not vanish in any point of the 2-adic circle T_{2}^{1}. In this sense, X_{f} is “2-adically expansive”. Consider the square root of −15 in ℤ_{2} given by the 2-adically convergent series

\sqrt{−15} = (1 + (−16))^{1/2} = ∑_{ν=0}^{∞} \binom{1/2}{ν}(−1)^{ν}2^{4ν} .

The zeroes of f(T) in \overline{ℚ}_{2} are given by α_{±} = \frac{1}{4}(1 ± \sqrt{−15}) ∈ ℚ_{2} where |α_{+}|_{2} = 2 and |α_{−}|_{2} = 1/2. Successive approximations for α_{+} coming from the series for \sqrt{−15} are 1/2, −3/2, −19/2, −83/2. By Theorem 9 and formula (6) the 2-adic entropy of X_{f} is given by

h_{2}(f) = log_{2} α_{+} ∈ ℤ_{2} .

Note that f viewed as a complex valued function has both its zeroes on S^{1}, so that X_{f} is not expansive in the usual sense. The topological entropy is h(f) = log 2.

We end this section by mentioning another example of a dynamical system where the p-adic entropy exists.

Proposition 12. For φ in GL_{d}(ℤ) without eigenvalues in \overline{ℚ}_{p} of p-adic absolute value one, the p-adic entropy of the φ-action on (ℝ/ℤ)^{d} in the sense of (2) exists and is zero.

Proof. We have

|Fix (φ^{n} | (ℝ/ℤ)^{d}| = |ℤ^{d}/(1 − φ^{n})(ℤ^{d})|
= ± det(1 − φ^{n}) = ± ∏_{λ}(1 − λ^{n}) .

Here the product runs over the eigenvalues λ ∈ \overline{ℚ}_{p} of φ. Thus we get

\frac{1}{n} log_{p} |Fix (φ^{n} | (ℝ/ℤ)^{d}| = ∑_{λ} \frac{1}{n} log_{p}(1 − λ^{n}) .  (17)

The characteristic polynomial of φ is monic with integer coefficients. Hence the eigenvalues λ are algebraic integers. In particular, we have |λ|_{p} ≤ 1 and hence |λ|_{p} < 1 since we assumed that |λ|_{p} ≠ 1.

Thus it suffices to show that for λ ∈ \overline{ℚ}_{p} with |λ|_{p} < 1 we have

lim_{n→∞} \frac{1}{n} log_{p}(1 − λ^{n}) = 0 .

This is a consequence of the following estimate:

|\frac{1}{n} log_{p}(1 − λ^{n})|_{p} = | ∑_{ν=1}^{∞} \frac{1}{νn}λ^{νn}|_{p} ≤ max_{ν} |\frac{1}{νn}λ^{νn}|_{p}
≤ max_{i≥n} |\frac{1}{i}λ^{i}|_{p} .

The last expression tends to zero for n → ∞ since |λ|_{p} < 1. ⊓ ⊔

4 The logarithm on the 1-units of a p-adic Banach algebra

For a discrete group Γ we would like to define a homomorphism

log_{p} det_{Γ} : c_{0}(Γ)^{∗} ⟶ ℚ_{p}

which should be a p-adic replacement for the map

log det_{N Γ} : L^{1}(Γ)^{∗} ⊂ (N Γ)^{∗} ⟶ ℝ .

More generally, we would like to define such a map on GL_{r}(c_{0}(Γ)). In this section we give the definition on the subgroup of 1-units and relate log_{p} det_{Γ} to p-adic entropy. The extension to a map on all of c_{0}(Γ)^{∗} will be done in the next section for suitable classes of groups Γ using rather deep facts about group rings.

Let B be a p-adic Banach algebra over ℚ_{p} whose norm ∥ ∥ takes values in p^{ℤ} ∪ {0}. A trace functional on B is a continuous linear map tr_{B} : B → ℚ_{p} which vanishes on commutators [a, b] = ab − ba of elements in B. For b ∈ B and c ∈ B^{∗} we have

tr_{B}(cbc^{−1}) = tr_{B}(b) .  (18)

Set A = B^{0} = {b ∈ B | ∥b∥ ≤ 1} and let U^{1} be the normal subgroup of 1-units in A^{∗}. The logarithmic series

log : U^{1} ⟶ A , log u = − ∑_{ν=1}^{∞} \frac{(1 − u)^{ν}}{ν}

converges and defines a continuous map. An argument with formal power series shows that we have

log uv = log u + log v  (19)

if the elements u and v in U^{1} commute with each other.

The next result is a consequence of the Campbell–Baker–Hausdorff formula.

Theorem 13. The map

tr_{B} log : U^{1} ⟶ ℤ_{p}

is a homomorphism. For u in U^{1} and a in A^{∗} we have

tr_{B} log(aua^{−1}) = tr_{B} log(u) .  (20)

Proof. Formula (20) follows from (18). From [Bou72] Ch. II, § 8 we get the following information about log. Set G = \{b ∈ B | ∥b∥ < p^{− \frac{1}{p−1}} \}. Then the exponential series defines a bijection exp : G ⟶^{∼} 1 + G with inverse exp^{−1} = log |_{1+G}. For x, y in G we have

exp x · exp y = exp h(x, y)  (21)

where h(x, y) ∈ G is given by a convergent series in B. It has the form

h(x, y) = x + y + series of (iterated) commutators .

Elements u, v of 1 + G have the form u = exp x and v = exp y. Taking the log of relation (21) and applying tr_{B} we get

tr_{B} log(uv) = tr_{B}h(x, y) = tr_{B}(x + y)
= tr_{B} log u + tr_{B} log v .  (22)

Hence tr_{B} log is a homomorphism on the subgroup 1+G of U^{1}. By assumption the norm of B takes values in p^{ℤ} ∪{0}. For p ≠ 2 we therefore have 1+G = U^{1} and we are done. For p = 2 the restriction of the map

φ = tr_{B} log : U^{1} → ℚ_{2}

to 1 + G = 1 + 4A is a homomorphism by (22). We have to show that it is a homomorphism on U^{1} = 1 + 2A as well. For u in U^{1} we have φ(u) = \frac{1}{2}φ(u^{2}) by (19) and u^{2} lies in 1+4A. Now consider elements u, v in U^{1}. Then we have

φ(uv) = \frac{1}{2}φ((uv)^{2}) = \frac{1}{2}φ(uvuv) =^{(20)} \frac{1}{2}φ(u^{2}vuvu^{−1})
= \frac{1}{2}φ(u^{2}) + \frac{1}{2}φ(vuvu^{−1})

since u^{2} and vuvu^{−1} lie in 1 + 4A where φ is a homomorphism. By similar arguments we get

φ(uv) = φ(u) + \frac{1}{2}φ(v^{2}uvu^{−1}v^{−1})
= φ(u) + φ(v) + \frac{1}{2}φ(uvu^{−1}v^{−1}) .

Thus we must show that φ(uvu^{−1}v^{−1}) = 0. By (20) we have φ(uvu^{−1}v^{−1}) = φ(vu^{−1}v^{−1}u) and hence using that both uvu^{−1}v^{−1} and vu^{−1}v^{−1}u lie in 1+4A we find

2φ(uvu^{−1}v^{−1}) = φ(uvu^{−1}v^{−1}) + φ(vu^{−1}v^{−1}u)
= φ(uvu^{−2}v^{−1}u) =^{(20)} φ(u^{−2}v^{−1}u^{2}v)
= φ(u^{−2}) + φ(v^{−1}u^{2}v)
=^{(20)} φ(u^{−2}) + φ(u^{2}) = φ(e)
= 0 .
⊓ ⊔

For a discrete group Γ, the map

tr_{Γ} : c_{0}(Γ) ⟶ ℚ_{p} , tr_{Γ} (∑a_{γ}γ) = a_{e}

defines a trace functional on c_{0}(Γ). Let B = M_{r}(c_{0}(Γ)) be the p-adic Banach algebra over ℚ_{p} of r × r-matrices (a_{ij}) with entries in c_{0}(Γ) and equipped with the norm ∥(a_{ij})∥ = max_{ij} ∥a_{ij}∥. The composition:

tr_{Γ} : M_{r}(c_{0}(Γ)) ⟶^{tr} c_{0}(Γ) ⟶^{tr_{Γ}} ℚ_{p}

defines a trace functional on M_{r}(c_{0}(Γ)).

The algebra A = B^{0} is given by M_{r}(c_{0}(Γ, ℤ_{p})) and we have U^{1} = 1 + pM_{r}(c_{0}(Γ, ℤ_{p})). The exact sequence (15) becomes the exact sequence of groups:

1 ⟶ 1 + pM_{r}(c_{0}(Γ, ℤ_{p})) ⟶ GL_{r}(c_{0}(Γ, ℤ_{p})) ⟶ GL_{r}(𝔽_{p}Γ) ⟶ 1 . (23)

According to theorem 13 the map

log_{p} det_{Γ} := tr_{Γ} log : 1 + pM_{r}(c_{0}(Γ, ℤ_{p})) ⟶ ℤ_{p}  (24)

is a homomorphism of groups.

Example 14. For Γ = ℤ^{d}, in the notation of example 5 we have a commutative diagram, c.f. [BD99] Lemma 1.1

[figure: c_{0}(Γ); ∼; _{tr_{Γ}}; ℚ_{p}⟨t_{1}^{±1}, …, t_{d}^{±1}⟩; _{∫_{T^{d}_{p}}}; ℚ_{p}; ℚ_{p} .]

It follows that for a 1-unit f in M_{r}(c_{0}(Γ)) we have:

log_{p} det_{Γ} f = ∫_{T_{p}^{d}} log det f = m_{p}(det f) .  (25)

Here we have used the relation

tr log f = log det f in c_{0}(Γ) ,  (26)

where det : GL_{r}(c_{0}(Γ)) → c_{0}(Γ)^{∗} is the determinant and tr the trace for matrices over the commutative ring c_{0}(Γ). Note that det maps 1-units to 1units. Relation (26) can be proved by embedding the integral domain c_{0}(Γ) = ℚ_{p}⟨t_{1}^{±1}, …, t_{d}^{±1}⟩ into its quotient field and applying [Har77] Appendix C, Lemma 4.1.

For finite groups Γ the map log_{p} det_{Γ} can be calculated as follows. For f in M_{r}(c_{0}(Γ)) = M_{r}(ℚ_{p}Γ) let ρ_{f} be the endomorphism of (ℚ_{p}Γ)^{r} by right multiplication with f^{∗} and det_{ℚ_{p}}(ρ_{f}) its determinant over ℚ_{p}.

Proposition 15. Let Γ be finite. Then we have

log_{p} det_{Γ} f = \frac{1}{|Γ |} log_{p} det_{ℚ_{p}}(ρ_{f})  (27)

for f in 1 + pM_{r}(ℤ_{p}Γ).

Remark 16. Since ρ_{fg} = ρ_{f}ρ_{g}, the equation in the proposition shows that log_{p} det_{Γ} is a homomorphism – something we know in general by theorem 13. For finite Γ the group GL_{r}(𝔽_{p}Γ) is finite. Hence, by (23) there is at most one way to extend log_{p} det_{Γ} from 1 + pM_{r}(ℤ_{p}Γ) to a homomorphism from GL_{r}(ℤ_{p}Γ) to ℚ_{p}. Namely, we have to set

log_{p} det_{Γ} f := \frac{1}{N} log_{p} det_{Γ} f^{N} ,

where N ≥ 1 is any integer with \overline{f}^{N} = 1 in GL_{r}(𝔽_{p}Γ). Because of (19) this is well defined but it is not clear from the definition that we get a homomorphism. However, for the same f, N we have

log_{p} det_{ℚ_{p}}(ρ_{f}) = \frac{1}{N} log_{p} det_{ℚ_{p}}(ρ_{f^{N}}) .

Hence equation (27) holds for all f in GL_{r}(ℤ_{p}Γ) and it follows that log_{p} det_{Γ} extends to a homomorphism on GL_{r}(ℤ_{p}Γ). In the next section such arguments will be generalized to infinite groups with the help of K-theory.

Proof of Proposition 15. Under the continuous homomorphism of p-adic Banach algebras over ℤ_{p}

ρ : M_{r}(ℤ_{p}Γ) ⟶ End_{ℤ_{p}}(ℤ_{p}Γ)^{r}

the groups of 1-units are mapped to each other. Hence we have

log ρ_{f} = ρ_{log f}  (28)

for f in 1 + pM_{r}(ℤ_{p}Γ).

On the other hand we have

tr_{Γ} (g) = \frac{1}{|Γ |}tr(ρ_{g})  (29)

for any element g of M_{r}(ℚ_{p}Γ). This is proved first for r = 1 by checking the cases where g = γ is an element of Γ. Then one extends to arbitrary r by thinking of ρ_{g} as a block matrix with blocks of size |Γ | × |Γ |.

Combining (28) and (29) we find:

log_{p} det_{Γ} f = tr_{Γ} log f
= \frac{1}{|Γ |}tr(ρ_{log f})
= \frac{1}{|Γ |}tr(log ρ_{f})
= \frac{1}{|Γ |} log_{p} det_{ℚ_{p}}(ρ_{f}) .

The last equation is proved by writing ρ_{f} in triangular form in a suitable basis over \overline{ℚ}_{p} and observing that the eigenvalues of ρ_{f} are 1-units in \overline{ℚ}_{p}. □

The next result is necessary to prove the relation of log_{p} det_{Γ} f with p-adic entropies.

Proposition 17. Let Γ be a residually finite countable discrete group and Γ_{n} → e a sequence as in the introduction. For f in 1+pM_{r}(c_{0}(Γ, ℤ_{p})) consider its image f^{(n)} in 1+pM_{r}(ℤ_{p}Γ^{(n)}) where Γ^{(n)} is the finite group Γ^{(n)} = Γ/Γ_{n}. Then we have

log_{p} det_{Γ} f = lim_{n→∞} log_{p} det_{Γ^{(n)}}f^{(n)} in ℤ_{p} .

Proof. The algebra map M_{r}(c_{0}(Γ)) → M_{r}(c_{0}(Γ^{(n)})) sending f to f^{(n)} is continuous since we have ∥f^{(n)}∥ ≤ ∥f∥. For f in 1 + pM_{r}(c_{0}(Γ, ℤ_{p})) we therefore get:

(log f)^{(n)} = log f^{(n)} in M_{r}(c_{0}(Γ^{(n)})) .

The next claim for g = log f thus implies the assertion. ⊓ ⊔

Claim. For g in M_{r}(c_{0}(Γ)) we have

tr_{Γ} (g) = lim_{n→∞} tr_{Γ^{(n)}}(g^{(n)}) .

Proof. We may assume that r = 1. Writing g = ∑ a_{γ}γ with a_{γ} ∈ ℚ_{p}, |a_{γ}|_{p} → 0 for γ → ∞ we have

|tr_{Γ} (g) − tr_{Γ^{(n)}}(g^{(n)})|_{p} = |a_{e} − ∑_{\overline{γ}=e} a_{γ} |_{p}
= | ∑_{γ∈Γ_{n}\e} a_{γ} |_{p}
≤ max_{γ∈Γ_{n}\e} |a_{γ}|_{p} .

For ε > 0 there is a finite subset S = S_{ε} of Γ such that |a_{γ}|_{p} < ε for γ ∈ Γ \S. Only e is contained in infinitely many Γ_{n}’s. Hence there is some n_{0} such that (Γ_{n} \ e) ∩ S = ∅ i.e. Γ_{n} \ e ⊂ Γ \ S for all n ≥ n_{0}. It follows that for n ≥ n_{0} we have |tr_{Γ} (g) − tr_{Γ^{(n)}}(g^{(n)})|_{p} ≤ ε. ⊓ ⊔ Corollary 18. Let Γ be a residually finite countable discrete group and f an element of M_{r}(ℤΓ) which is a 1-unit in M_{r}(c_{0}(Γ)). Then the p-adic entropy h_{p}(f) of the Γ -action on X_{f} exists for all Γ_{n} → e and we have

h_{p}(f) = log_{p} det_{Γ} f in ℤ_{p} .

Proof. By propositions 7 and 15 we have

\frac{1}{(Γ : Γ_{n})} log_{p} |Fix_{Γ_{n}}(X_{f})| = \frac{1}{(Γ : Γ_{n})} log_{p} det_{ℚ_{p}}(ρ_{f^{(n)}})
= log_{p} det_{Γ^{(n)}}(f^{(n)}) .

Hence the assertion follows from Proposition 17. ⊓ ⊔

5 A p-adic logarithmic Fuglede–Kadison determinant and its relation to p-adic entropy

Having defined a homomorphism log_{p} det_{Γ} on 1+pc_{0}(Γ, ℤ_{p}) in (24) one would like to use the exact sequence (16) to extend it to c_{0}(Γ, ℤ_{p})^{∗}. However, for infinite groups Γ the abelianization of the group 𝔽_{p}[Γ]^{∗} divided by the image of Γ is not known to be torsion in any generality – as far as I know. However, corresponding results are known for K_{1} of 𝔽_{p}[Γ] and this determines our approach which even for r = 1 requires the preceeding considerations for matrix algebras.

For a unital ring R recall the embedding GL_{r}(R) ↪ GL_{r+1}(R) mapping a to ( \begin{matrix}a & 0 \\ 0 & 1 \end{matrix} ). Let GL_{∞}(R) be the union of the GL_{r}(R)’s. We will view elements of GL_{∞}(R) as infinite matrices with 1’s on the diagonal and only finitely many further nonzero entries. The subgroup E_{r}(R) ⊂ GL_{r}(R) of elementary matrices is the subgroup generated by matrices which have 1’s on the diagonal and at most one further non-zero entry. Let E_{∞}(R) be their union and set K_{1}(R) = GL_{∞}(R)/E_{∞}(R). It is known that we have E_{∞}(R) = (GL_{∞}(R), GL_{∞}(R)) and hence that K_{1}(R) = GL_{∞}(R)^{ab} c.f.

[Mil71] § 3. The Whitehead group over 𝔽_{p} of a discrete group Γ is defined to be

W h^{𝔽_{p}}(Γ) := K_{1}(𝔽_{p}[Γ])/⟨Γ ⟩ .

Here ⟨Γ ⟩ is the image of Γ under the canonical map 𝔽_{p}[Γ]^{∗} → K_{1}(𝔽_{p}[Γ]).

We can treat groups for which W h^{𝔽_{p}}(Γ) is torsion. According to [FL03] Theorem 1.1 this is the case for torsion-free elementary amenable groups Γ. Recently, in [BLR] it has been shown for a larger class of groups that W h^{𝔽_{p}}(Γ) is torsion. Apart from the elementary amenable groups, this class comprises all word hyperbolic groups. It is closed under subgroups, finite products, colimits and suitable extensions.

Theorem 19. Let Γ be a countable discrete residually finite group such that W h^{𝔽_{p}}(Γ) is torsion. Then there is a unique homomorphism

log_{p} det_{Γ} : K_{1}(c_{0}(Γ, ℤ_{p})) ⟶ ℚ_{p}

with the following properties: a For every r ≥ 1 the composition

1 + pM_{r}(c_{0}(Γ, ℤ_{p})) ↪ GL_{r}(c_{0}(Γ, ℤ_{p})) → K_{1}(c_{0}(Γ, ℤ_{p})) ⟶^{log_{p} det_{Γ}} ℚ_{p}

coincides with the map log_{p} det_{Γ} introduced in (24). b On the image of Γ in K_{1}(c_{0}(Γ, ℤ_{p})) the map log_{p} det_{Γ} vanishes.

Proof. Set A = c_{0}(Γ, ℤ_{p}) and \overline{A} = A/pA = 𝔽_{p}[Γ]. The reduction map A → \overline{A} induces an exact sequence

0 → Γ E_{∞}(A)(1 + pM_{∞}(A))/Γ E_{∞}(A) → K_{1}(A)/⟨Γ ⟩ → K_{1}(\overline{A})/⟨Γ ⟩ . (30)

Here M_{∞}(A) is the (non-unital) algebra of infinite matrices (a_{ij})_{i,j≥1} with only finitely many non-zero entries. Note that 1 + pM_{∞}(A) is a subgroup of GL_{∞}(A) since 1 + pM_{r}(A) is a subgroup of GL_{r}(A). Moreover Γ E_{∞}(A) is a normal subgroup of GL_{∞}(A). Hence the sequence (30) becomes an exact sequence:

0 → (1 + pM_{∞}(A))/Γ E_{∞}(A) ∩ (1 + pM_{∞}(A)) → K_{1}(A)/⟨Γ ⟩ → K_{1}(\overline{A})/⟨Γ ⟩ .

(31)

Since ℚ_{p} is uniquely divisible this implies the uniqueness assertion in the theorem for any group Γ such that W h^{𝔽_{p}}(Γ) = K_{1}(\overline{A})/⟨Γ ⟩ is torsion. For the existence, we first note that the homomorphisms defined in (24) induce a homomorphism

log_{p} det_{Γ} : 1 + pM_{∞}(A) ⟶ ℤ_{p} .

We have to show that log_{p} det_{Γ} f = 0 for every f in 1+pM_{∞}(A) which also lies in Γ E_{∞}(A). Under our identification of GL_{r}(A) with a subgroup of GL_{∞}(A) we find some r ≥ 1 such that we have

f = i(γ)e_{1} ⋯ e_{N} in 1 + pM_{r}(A) .

Here the e_{i} are elementary r × r-matrices and i(γ) = ( \begin{matrix}γ & 0 \\ 0 & 1_{r−1} \end{matrix} ) for some γ in Γ.

According to Propositions 15 and 17 we have for any choice of sequence Γ_{n} → e:

log_{p} det_{Γ} f = lim_{n→∞} \frac{1}{(Γ : Γ_{n})} log_{p} det_{ℚ_{p}}(ρ_{f^{(n)}}) .

On the other hand:

det_{ℚ_{p}}(ρ_{f^{(n)}}) = det_{ℚ_{p}}(ρ_{i(γ)^{(n)}e_{1}^{(n)}...e_{N}^{(n)}}) = det_{ℚ_{p}}(ρ_{i(γ)^{(n)}}) ∏_{i} det_{ℚ_{p}}(ρ_{e_{i}^{(n)}}) .

Let 𝔟 be a basis of ℚ_{p}[Γ^{(n)}]. In the basis (𝔟, …, 𝔟) of ℚ_{p}[Γ^{(n)}]^{r} the endomorphism ρ_{e_{i}^{(n)}} is given by a matrix of |Γ^{(n)}| × |Γ^{(n)}|-blocks. The diagonal blocks are identity matrices. At most one of the other blocks is non-zero. In particular, the matrix is triangular and we have det_{ℚ_{p}}(ρ_{e_{i}^{(n)}}) = 1. In the same basis ρ_{i(γ)^{(n)}} is a permutation matrix and hence det_{ℚ_{p}}(ρ_{i(γ)^{(n)}}) = ±1. It follows that we have log_{p} det_{Γ} f = 0 as we wanted to show. ⊓ ⊔

I think that Theorem 19 should also hold without the condition that Γ is residually finite.

Remark 20. For Γ = ℤ^{d} and f in GL_{r}(c_{0}(Γ, ℤ_{p})), writing [f] for the class of f in K_{1}, we have

log_{p} det_{Γ} [f] = m_{p}(det f)

extending equation (25).

This follows from the uniqueness assertion in Theorem 19. Namely, the map [f] ↦ m_{p}(det f) defines a homomorphism on K_{1} which according to equation (25) satisfies condition a. It satisfies condition b as well, since log_{p} vanishes on roots of unity and hence we have m_{p}(t^{ν}) = 0 for all ν in ℤ^{d}, c.f. Example 5.

Definition 21. For any group Γ as in the theorem we define the homomorphism log_{p} det_{Γ} on GL_{r}(c_{0}(Γ, ℤ_{p})) to be the composition

log_{p} det_{Γ} : GL_{r}(c_{0}(Γ, ℤ_{p})) ⟶ K_{1}(c_{0}(Γ, ℤ_{p})) ⟶^{log_{p} det_{Γ}} ℚ_{p} .

If we unravel the definitions we get the following description of this map. Given a matrix f in GL_{r}(c_{0}(Γ, ℤ_{p})) there are integers N ≥ 1 and s ≥ r such that in GL_{s}(c_{0}(Γ, ℤ_{p})) we have f^{N} = i(γ)εg with ε in E_{s}(c_{0}(Γ, ℤ_{p})), g in 1 + pM_{s}(c_{0}(Γ, ℤ_{p})) and i(γ) the s × s-matrix ( \begin{matrix}γ & 0 \\ 0 & 1_{s−1} \end{matrix} ). Then we have

log_{p} det_{Γ} f = \frac{1}{N} log_{p} det_{Γ} g = \frac{1}{N} tr_{Γ} log g .  (32)

We can now prove the following extension of Corollary 18.

Theorem 22. Let Γ be a residually finite countable discrete group such that W h^{𝔽_{p}}(Γ) is torsion. Let f be an element of M_{r}(ℤΓ) ∩ GL_{r}(c_{0}(Γ, ℤ_{p})). Then h_{p}(f) exists for all Γ_{n} → e and we have

h_{p}(f) = log_{p} det_{Γ} f in ℚ_{p} .

Proof. Let us write f^{N} = i(γ)εg as above. Then by Proposition 7 we have

log_{p} |Fix_{Γ_{n}}(X_{f})| = log_{p} det_{ℚ_{p}}(ρ_{f^{(n)}})
= \frac{1}{N} log_{p} det_{ℚ_{p}}(ρ_{i(γ)^{(n)}}) + \frac{1}{N} log_{p} det_{ℚ_{p}}(ρ_{ε^{(n)}})

+\frac{1}{N} log_{p} det_{ℚ_{p}}(ρ_{g^{(n)}}) .

Note here that the composition

M_{s}(c_{0}(Γ)) ⟶ M_{s}(c_{0}(Γ^{(n)})) ⟶^{ρ} End_{ℚ_{p}}(ℚ_{p}Γ^{(n)})^{s}

is a homomorphism of algebras.

As in the proof of Theorem 19 we see that the terms log_{p} det_{ℚ_{p}}(ρ_{i(γ)^{(n)}}) and log_{p} det_{ℚ_{p}}(ρ_{ε^{(n)}}) vanish. This gives

\frac{1}{(Γ : Γ_{n})} log_{p} |Fix_{Γ_{n}}(X_{f})| = \frac{1}{(Γ : Γ_{n})} \frac{1}{N} log_{p} det_{ℚ_{p}}(ρ_{g^{(n)}})
= \frac{1}{N} log_{p} det_{Γ^{(n)}}(g^{(n)}) by Proposition 15.

Using Proposition 17 we get in the limit n → ∞ that

h_{p}(f) = \frac{1}{N} log_{p} det_{Γ} (g) =^{(32)} log_{p} det f .
⊓ ⊔

For groups Γ as in Theorem 22 whose group ring 𝔽_{p}Γ has no zero divisors it is possible to extend the definition of log_{p} det_{Γ} from c_{0}(Γ, ℤ_{p})^{∗} to c_{0}(Γ)^{∗}. Namely, by Proposition 4 we know that

c_{0}(Γ)^{∗} = p^{ℤ}c_{0}(Γ, ℤ_{p})^{∗} and p^{ℤ} ∩ c_{0}(Γ, ℤ_{p})^{∗} = 1 .

Hence there is a unique homomorphism

log_{p} det_{Γ} : c_{0}(Γ)^{∗} ⟶ ℚ_{p}

which agrees with log_{p} det_{Γ} previously defined on c_{0}(Γ, ℤ_{p})^{∗} in Definition 21 and satisfies

log_{p} det_{Γ} (p) = 0 .

Let Γ be a torsion-free elementary amenable group. Then according to [KLM88] Theorem 1.4 the group ring 𝔽_{p}Γ has no zero divisors and according to [FL03] Theorem 1.1 the group W h^{𝔽_{p}}(Γ) is torsion. Hence log_{p} det_{Γ} is defined on c_{0}(Γ)^{∗} and this is the map used in Theorem 2. Proof of Theorem 2. Writing f in ℤΓ ∩ c_{0}(Γ)^{∗} as a product f = p^{ν}g with g in c_{0}(Γ, ℤ_{p})^{∗} it follows that g ∈ ℤΓ and Proposition 7 shows that we have

log_{p} |Fix_{Γ_{n}}(X_{f})| = log_{p} det_{ℚ_{p}}(ρ_{f^{(n)}})
= log_{p} det_{ℚ_{p}}(ρ_{g^{(n)}})
= log_{p} |Fix_{Γ_{n}}(X_{g})| .

Note here that we have log_{p}(p) = 0. It follows from Theorem 22 applied to g that for all Γ_{n} → e we get:

h_{p}(f) = h_{p}(g) = log_{p} det_{Γ} g = log_{p} det_{Γ} f .

□

For Γ = ℤ^{d} it follows from Remark 20 that for any f in c_{0}(ℤ^{d})^{∗} = ℚ_{p}⟨t_{1}^{±1}, …, t_{d}^{±1}⟩^{∗} we have:

log_{p} det_{Γ} f = m_{p}(f) .

Hence Theorem 1 is a special case of Theorem 2.

Concerning approximations of log_{p} det_{Γ} f we note that Proposition 17 extends to more general cases.

Proposition 23. Let Γ be a residually finite countable discrete group and Γ_{n} → e as in the introduction. For f in M_{r}(c_{0}(Γ)) let f^{(n)} be its image in M_{r}(ℚ_{p}Γ^{(n)}). Then the formula

log_{p} det_{Γ} f = lim_{n→∞} \frac{1}{(Γ : Γ_{n})} log_{p} det_{ℚ_{p}}(ρ_{f^{(n)}})  (33)

holds whenever log_{p} det_{Γ} f is defined. These are the cases a where f is in 1 + pM_{r}(c_{0}(Γ, ℤ_{p})) b where W h^{𝔽_{p}}(Γ) is torsion and f is in GL_{r}(c_{0}(Γ, ℤ_{p})) c where W h^{𝔽_{p}}(Γ) is torsion, 𝔽_{p}Γ has no zero divisors and f is in c_{0}(Γ)^{∗}.

Proof. The assertions follow from Propositions 15 and 17 together with calculations as in the proofs of theorems 19 and 22. ⊓ ⊔

We end the paper with some open questions: Is there a dynamical criterion for the existence of the limit defining p-adic entropy? Is there a notion of “padic expansiveness” for Γ -actions on compact spaces X which for the systems X_{f} with f in M_{r}(ℤΓ) translates into the condition that f is invertible in M_{r}(c_{0}(Γ)), (or in M_{r}(c_{0}(Γ, ℤ_{p})))? In fact, I assume that p-adic entropy can only be defined for “p-adically expansive” systems, c.f. [BD99] Remark after proposition 1.3. What is the dynamical meaning of Proposition 12? Is there a direct proof that the limit in formula (33) exists?

Finally, in [BD99] a second version of a p-adic Mahler measure was defined which involves both the p-adic and the archimedian valuations of ℚ. Can this be obtained for the systems X_{f} by doing something more involved with the fixed points than taking their cardinalities and forming the limit (3)?

References

[BD99] A. Besser, C. Deninger p-adic mahler measures, J. reine angew. Math., 517:19–50, 1999.

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