-algebras from self-similar actions

New York Journal of Mathematics

New York J. Math.27(2021) 923–942.

On groupoids and 𝑪

-algebras from self-similar actions

Valentin Deaconu

Abstract. Given a self-similar groupoid action(𝐺, 𝐸)on the path space of a finite graph, we study the associated Exel-Pardo étale groupoid𝒢(𝐺, 𝐸)and its𝐶^∗-algebra𝐶^∗(𝐺, 𝐸). We review some facts about groupoid actions, skew products and semi-direct products and generalize a result of Renault about similarity of groupoids which resembles Takai duality. We also describe a general strategy to compute the𝐾-theory of𝐶^∗(𝐺, 𝐸)and the homology of 𝒢(𝐺, 𝐸)in certain cases and illustrate with an example.

Contents

1. Introduction 923

2. Homology of étale groupoids 924

3. Groupoid actions and similarity 926

4. Self-similar groupoid actions and their𝐶^∗-algebra 931 5. Exel-Pardo groupoids for self-similar actions 935

6. Example 937

References 941

1. Introduction

Informally, a self-similar action is given by isomorphisms between parts of an object (think fractals or Julia sets) at different scales. Self-similar actions were studied intensely after exotic examples of groups acting on rooted trees and generated by finite automata, like infinite residually finite torsion groups, and groups of intermediate growth were constructed by Grigorchuk in the 1980’s.

Using the Pimsner construction from a𝐶^∗-correspondence, Nekrashevych in- troduced the𝐶^∗-algebras associated with self-similar group actions in [13,14], where important results about their structure and their𝐾-theory were obtained.

Received December 29, 2020.

2020Mathematics Subject Classification. Primary 46L05.

Key words and phrases. Homology of groupoids; self-similar action; similarity of groupoids;

Cuntz-Pimsner algebra; K-theory.

ISSN 1076-9803/2021

923

Motivated by the construction of all Kirchberg algebras in the UCT class us- ing topological graphs given by Katsura, in [7] Exel and Pardo introduced self- similar group actions on graphs and realized their𝐶^∗-algebras as groupoid𝐶^∗- algebras.

In this paper, we are interested in self-similar actions of groupoids𝐺on the path space of finite directed graphs𝐸as introduced and studied in [10], where the main goal was to find KMS states on some resulting dynamical systems. We generalize certain results of Exel and Pardo, in particular we define a groupoid 𝒢(𝐺, 𝐸)and discuss the structure of the𝐶^∗-algebra𝐶^∗(𝐺, 𝐸), defined as a Cuntz- Pimsner algebra of a𝐶^∗-correspondence over𝐶^∗(𝐺). The𝐶^∗-algebra𝐶^∗(𝐺, 𝐸) has a natural gauge action and contains copies of𝐶^∗(𝐸)and𝐶^∗(𝐺). In general, its structure is rather intricate; in a particular case,𝐶^∗(𝐺, 𝐸) ≅ 𝐶^∗(𝐸) ⋊ 𝐺.

We begin with a review of étale groupoid homology, then we recall some facts about groupoid actions, skew products and semi-direct products and generalize a result of Renault about similarity of groupoids in the spirit of Takai duality.

We also describe a general strategy to compute the𝐾-theory of𝐶^∗(𝐺, 𝐸)and the homology of𝒢(𝐺, 𝐸)in certain cases. We illustrate with an example.

We expect that many of our results will be true for self-similar actions of groupoids on the path space of infinite graphs. For the case when𝐺is a group, see [8,11].

2. Homology of étale groupoids

A groupoid𝐺is a small category with inverses. The set of objects is denoted by𝐺⁽⁰⁾. We will use𝑑and𝑡for the domain and terminus maps𝑑, 𝑡 ∶ 𝐺 → 𝐺⁽⁰⁾ to distinguish them from the range and source maps𝑟, 𝑠on directed graphs. For 𝑢, 𝑣 ∈ 𝐺⁽⁰⁾, we write

𝐺_𝑢= {𝑔 ∈ 𝐺 ∶ 𝑑(𝑔) = 𝑢}, 𝐺^𝑣 = {𝑔 ∈ 𝐺 ∶ 𝑡(𝑔) = 𝑣}, 𝐺_𝑢^𝑣 = 𝐺_𝑢∩ 𝐺^𝑣. The set of composable pairs is denoted𝐺⁽²⁾.

An étale groupoid is a topological groupoid where the terminus map𝑡(and necessarily the domain map𝑑) is a local homeomorphism (as a map from𝐺to 𝐺). The unit space𝐺⁽⁰⁾of an étale groupoid is always an open subset of𝐺. Definition 2.1. Let𝐺be an étale groupoid. A bisection is an open subset𝑈 ⊆ 𝐺such that𝑑and𝑡are both injective when restricted to𝑈.

Two units𝑥, 𝑦 ∈ 𝐺⁽⁰⁾belong to the same𝐺-orbit if there exists𝑔 ∈ 𝐺such that𝑑(𝑔) = 𝑥and𝑡(𝑔) = 𝑦. We denote by orb𝐺(𝑥)the𝐺-orbit of𝑥. When every 𝐺-orbit is dense in𝐺⁽⁰⁾, the groupoid𝐺is called minimal. An open set𝑉 ⊆ 𝐺⁽⁰⁾ is called𝐺-full if for every𝑥 ∈ 𝐺⁽⁰⁾we have orb𝐺(𝑥) ∩ 𝑉 ≠ ∅. We denote by𝐺_𝑉 the subgroupoid{𝑔 ∈ 𝐺 | 𝑑(𝑔), 𝑡(𝑔) ∈ 𝑉}, called the restriction of𝐺to𝑉. When 𝐺is étale, the restriction𝐺_𝑉 is an open étale subgroupoid with unit space𝑉.

The isotropy group of a unit𝑥 ∈ 𝐺⁽⁰⁾is the group 𝐺_𝑥^𝑥∶= {𝑔 ∈ 𝐺 | 𝑑(𝑔) = 𝑡(𝑔) = 𝑥},

and the isotropy bundle is

𝐺^′∶= {𝑔 ∈ 𝐺 | 𝑑(𝑔) = 𝑡(𝑔)} =

⋃

𝑥∈𝐺⁽⁰⁾

𝐺_𝑥^𝑥.

A groupoid𝐺is said to be principal if all isotropy groups are trivial, or equiva- lently,𝐺^′= 𝐺⁽⁰⁾. We say that𝐺is effective if the interior of𝐺^′equals𝐺⁽⁰⁾. Definition 2.2. A second countable groupoid𝐺is ample if it is étale and𝐺⁽⁰⁾ is zero-dimensional; equivalently,𝐺 is ample if it has a basis of compact open bisections. An ample groupoid𝐺is elementary if it is compact and principal.

An ample groupoid is an𝐴𝐹groupoid if there exists an ascending chain of open elementary subgroupoids𝐺₁⊆ 𝐺₂⊆ ... ⊆ 𝐺such that𝐺 =⋃∞

𝑖=1𝐺_𝑖.

We recall now the definion of homology of étale groupoids which was in- troduced by Crainic and Moerdijk in [3]. Let𝐴be an Abelian group and let 𝜋 ∶ 𝑋 → 𝑌be a local homeomorphism between two locally compact Haus- dorff spaces. Given any𝑓 ∈ 𝐶_𝑐(𝑋, 𝐴), we define

𝜋_∗(𝑓)(𝑦) ∶= ∑

𝜋(𝑥)=𝑦

𝑓(𝑥).

It follows that𝜋_∗(𝑓) ∈ 𝐶_𝑐(𝑌, 𝐴). Given an étale groupoid𝐺, let𝐺⁽¹⁾ = 𝐺and for𝑛 ≥ 2let𝐺^(𝑛)be the space of composable strings of𝑛elements in𝐺with the product topology. For𝑛 ≥ 2and𝑖 = 0, ..., 𝑛, we let𝜕_𝑖 ∶ 𝐺^(𝑛) → 𝐺^(𝑛−1)be the face maps defined by

𝜕_𝑖(𝑔₁, 𝑔₂, ..., 𝑔_𝑛) =

⎧

⎨

⎩

(𝑔₂, 𝑔₃, ..., 𝑔_𝑛) if𝑖 = 0,

(𝑔₁, ..., 𝑔_𝑖𝑔_𝑖+1, ..., 𝑔_𝑛) if1 ≤ 𝑖 ≤ 𝑛 − 1, (𝑔₁, 𝑔₂, ..., 𝑔_𝑛−1) if𝑖 = 𝑛.

We define the homomorphisms𝛿_𝑛∶ 𝐶_𝑐(𝐺^(𝑛), 𝐴) → 𝐶_𝑐(𝐺^(𝑛−1), 𝐴)given by 𝛿₁= 𝑑_∗− 𝑡_∗, 𝛿_𝑛=

∑𝑛 𝑖=0

(−1)^𝑖𝜕_𝑖∗for𝑛 ≥ 2.

It can be verified that𝛿_𝑛◦𝛿_𝑛+1= 0for all𝑛 ≥ 1.

The homology groups𝐻_𝑛(𝐺, 𝐴)are by definition the homology groups of the chain complex𝐶_𝑐(𝐺^(∗), 𝐴)given by

0⟵ 𝐶^𝛿0 _𝑐(𝐺⁽⁰⁾, 𝐴)⟵ 𝐶^𝛿1 _𝑐(𝐺⁽¹⁾, 𝐴)⟵ 𝐶^𝛿2 _𝑐(𝐺⁽²⁾, 𝐴) ⟵ ⋯ ,

i.e. 𝐻_𝑛(𝐺, 𝐴) = ker 𝛿_𝑛∕im𝛿_𝑛+1, where𝛿₀ = 0. If𝐴 = ℤ, we write𝐻_𝑛(𝐺)for 𝐻_𝑛(𝐺, ℤ).

The following HK-conjecture of Matui states that the homology of an étale groupoid refines the𝐾-theory of the reduced groupoid𝐶^∗-algebra. Let𝐺 be a minimal effective ample Hausdorff groupoid with compact unit space. Then

𝐾_𝑖(𝐶_𝑟^∗(𝐺)) ≅

⨁∞ 𝑛=0

𝐻_2𝑛+𝑖(𝐺), for𝑖 = 0, 1.

Recently, this conjecture was the source of intense research. It was con- firmed for several groupoids like𝐴𝐹-groupoids, transformation groupoids of Cantor minimal systems, groupoids of shifts of finite type and products of group- oids of shifts of finite type (see [12]). The homology of ample Hausdorff group- oids was investigated in [9], with emphasis on the Renault-Deaconu groupoids associated to𝑘pairwise-commuting local homeomorphisms of a zero dimen- sional space. It was shown that the homology of𝑘-graph groupoids can be com- puted in terms of the adjacency matrices, using spectral sequences and a chain complex developed by Evans in [6]. The HK-conjecture was also confirmed for groupoids on one-dimensional solenoids in [20]. Recently, counterexamples to the HK-conjecture of Matui were found by Scarparo in [18] and by Ortega and Sanchez in [16].

3. Groupoid actions and similarity

We recall the concept of a groupoid action on another groupoid from [1], page 122 and from [5].

Definition 3.1. A topological groupoid𝐺acts (on the right) on another topo- logical groupoid𝐻if there are a continuous open surjection𝑝 ∶ 𝐻 → 𝐺⁽⁰⁾and a continuous map𝐻 ∗ 𝐺 → 𝐻, write(ℎ, 𝑔) ↦ ℎ ⋅ 𝑔where

𝐻 ∗ 𝐺 = {(ℎ, 𝑔) ∈ 𝐻 × 𝐺 ∣ 𝑡(𝑔) = 𝑝(ℎ)}

such that

i)𝑝(ℎ ⋅ 𝑔) = 𝑑(𝑔)for all(ℎ, 𝑔) ∈ 𝐻 ∗ 𝐺,

ii)(ℎ, 𝑔₁) ∈ 𝐻 ∗ 𝐺and(𝑔₁, 𝑔₂) ∈ 𝐺⁽²⁾implies that(ℎ, 𝑔₁𝑔₂) ∈ 𝐻 ∗ 𝐺and ℎ ⋅ (𝑔₁𝑔₂) = (ℎ ⋅ 𝑔₁) ⋅ 𝑔₂,

iii)(ℎ₁, ℎ₂) ∈ 𝐻⁽²⁾ and(ℎ₁ℎ₂, 𝑔) ∈ 𝐻 ∗ 𝐺implies(ℎ₁, 𝑔), (ℎ₂, 𝑔) ∈ 𝐻 ∗ 𝐺 and

(ℎ₁ℎ₂) ⋅ 𝑔 = (ℎ₁⋅ 𝑔)(ℎ₂⋅ 𝑔), iv)ℎ ⋅ 𝑝(ℎ) = ℎfor allℎ ∈ 𝐻.

The action is called free ifℎ ⋅ 𝑔 = ℎimplies𝑔 = 𝑝(ℎ)and transitive if for all ℎ₁, ℎ₂∈ 𝐻there is𝑔 ∈ 𝐺withℎ₂= ℎ₁⋅ 𝑔.

Note that if𝐺acts on𝐻 on the right, we can define a left action of𝐺on𝐻 by taking𝑔 ⋅ ℎ ∶= ℎ ⋅ 𝑔⁻¹and vice versa.

Example3.2. Given a topological groupoid𝐺, a𝐺-module in [19] is a topolog- ical groupoid𝐴with domain and terminus maps equal to𝑝 ∶ 𝐴 → 𝐺⁽⁰⁾such that𝐴^𝑥_𝑥is an abelian group for all𝑥 ∈ 𝐺⁽⁰⁾,𝐺acts on𝐴as a space and such that for each𝑔 ∈ 𝐺 the action map𝛼_𝑔 ∶ 𝐴_𝑑(𝑔) → 𝐴_𝑡(𝑔) is a group homomorphism.

In particular,𝐴can be a trivial group bundle𝐺⁽⁰⁾× 𝐷 for𝐷 an abelian group and𝛼_𝑔= 𝑖𝑑_𝐷for all𝑔 ∈ 𝐺.

Remark3.3. If𝐺 acts on the groupoid𝐻, then𝐺 acts on the unit space𝐻⁽⁰⁾ using the restriction𝑝₀∶= 𝑝|_𝐻⁽⁰⁾ ∶ 𝐻⁽⁰⁾→ 𝐺⁽⁰⁾and we have𝑝 = 𝑝₀◦𝑡 = 𝑝₀◦𝑑, where𝑑, 𝑡 ∶ 𝐻 → 𝐻⁽⁰⁾. In particular,

𝑝(ℎ⁻¹) = 𝑝₀(𝑡(ℎ⁻¹)) = 𝑝₀(𝑑(ℎ)) = 𝑝(ℎ).

Using the fact thatℎ = ℎ𝑑(ℎ) = 𝑡(ℎ)ℎ, it follows that ℎ ⋅ 𝑔 = (ℎ ⋅ 𝑔)(𝑑(ℎ) ⋅ 𝑔) = (𝑡(ℎ) ⋅ 𝑔)(ℎ ⋅ 𝑔), so we deduce that𝑑(ℎ ⋅ 𝑔) = 𝑑(ℎ) ⋅ 𝑔and𝑡(ℎ ⋅ 𝑔) = 𝑡(ℎ) ⋅ 𝑔.

Definition 3.4. If𝐺acts on𝐻, then the semi-direct product groupoid𝐻 ⋊ 𝐺, also called the action groupoid, is defined as follows. As a set,

𝐻 ⋊ 𝐺 = 𝐻 ∗ 𝐺 = {(ℎ, 𝑔) ∈ 𝐻 × 𝐺 ∣ 𝑡(𝑔) = 𝑝(ℎ)}

and the multiplication is given by

(ℎ, 𝑔)(ℎ^′⋅ 𝑔, 𝑔^′) = (ℎℎ^′, 𝑔𝑔^′), when𝑡(𝑔^′) = 𝑑(𝑔)and𝑑(ℎ) = 𝑡(ℎ^′).

In a semi-direct product, the inverse is given by (ℎ, 𝑔)⁻¹= (ℎ⁻¹⋅ 𝑔, 𝑔⁻¹) and we get

(ℎ, 𝑔)⁻¹(ℎ, 𝑔) = (ℎ⁻¹⋅ 𝑔, 𝑔⁻¹)(ℎ, 𝑔) = ((ℎ⁻¹⋅ 𝑔)(ℎ ⋅ 𝑔), 𝑑(𝑔)) = (𝑑(ℎ) ⋅ 𝑔, 𝑑(𝑔)), (ℎ, 𝑔)(ℎ, 𝑔)⁻¹= (ℎ, 𝑔)(ℎ⁻¹⋅ 𝑔, 𝑔⁻¹) = (𝑡(ℎ), 𝑡(𝑔)).

Since𝑑(𝑔) = 𝑝(𝑑(ℎ) ⋅ 𝑔)and𝑡(𝑔) = 𝑝(𝑡(ℎ)), the unit space of𝐻 ⋊ 𝐺can be identified with𝐻⁽⁰⁾and then we make identifications

𝑑(ℎ, 𝑔) ≡ 𝑑(ℎ) ⋅ 𝑔, 𝑡(𝑔, ℎ) ≡ 𝑡(ℎ).

There is a groupoid homomorphism

𝜋 ∶ 𝐻 ⋊ 𝐺 → 𝐺, 𝜋(ℎ, 𝑔) = 𝑔

with kernel𝜋⁻¹(𝐺⁽⁰⁾) = {(ℎ, 𝑝(ℎ)) ∣ ℎ ∈ 𝐻}isomorphic to𝐻.

Remark3.5. The notion of groupoid action on another groupoid includes the action of a groupoid on a space and the action of a group on another group by automorphisms. A particular situation is when𝐺₁, 𝐺₂are groupoids and𝑍 is a (𝐺₁, 𝐺₂)-space, i.e. 𝑍 is a left𝐺₁-space, a right𝐺₂-space and the actions commute. Then𝐺₁⋉ 𝑍is a right𝐺₂-groupoid and𝑍 ⋊ 𝐺₂is a left𝐺₁-groupoid.

Definition 3.6. Suppose now that𝐺, Γare étale groupoids and that𝜌 ∶ 𝐺 → Γ is a groupoid homomorphism, also called a cocycle. The skew product groupoid 𝐺 ×_𝜌Γis defined as the set of pairs(𝑔, 𝛾) ∈ 𝐺 × Γsuch that(𝛾, 𝜌(𝑔)) ∈ Γ⁽²⁾with multiplication

(𝑔, 𝛾)(𝑔^′, 𝛾𝜌(𝑔)) = (𝑔𝑔^′, 𝛾)if(𝑔, 𝑔^′) ∈ 𝐺⁽²⁾ and inverse

(𝑔, 𝛾)⁻¹= (𝑔⁻¹, 𝛾𝜌(𝑔)).

In a skew product, we have𝑑(𝑔, 𝛾) = (𝑑(𝑔), 𝛾𝜌(𝑔))and𝑡(𝑔, 𝛾) = (𝑡(𝑔), 𝛾). Its unit space is

𝐺⁽⁰⁾ ∗ Γ = {(𝑢, 𝛾) ∈ 𝐺⁽⁰⁾× Γ ∶ 𝜌(𝑢) = 𝑑(𝛾)}.

In particular, if𝐺, 𝐻 are étale groupoids and 𝐺acts on𝐻 on the right, for the groupoid homomorphism𝜋 ∶ 𝐻 ⋊ 𝐺 → 𝐺, 𝜋(ℎ, 𝑔) = 𝑔we can form the skew product(𝐻 ⋊ 𝐺) ×_𝜋𝐺 made of triples(ℎ, 𝑔, 𝑔^′) ∈ 𝐻 × 𝐺 × 𝐺such that 𝑝(ℎ) = 𝑡(𝑔)and(𝑔^′, 𝑔) ∈ 𝐺⁽²⁾, with unit space𝐻⁽⁰⁾∗ 𝐺and operations

(ℎ, 𝑔, 𝑔^′)(ℎ^′, 𝑔^′′, 𝑔^′𝑔) = (ℎ(ℎ^′⋅ 𝑔⁻¹), 𝑔𝑔^′′, 𝑔^′), (ℎ, 𝑔, 𝑔^′)⁻¹= (ℎ⁻¹⋅ 𝑔, 𝑔⁻¹, 𝑔^′𝑔).

Remark3.7. Given a groupoid homomorphism𝜌 ∶ 𝐺 → Γ, there is a left action

̂

𝜌ofΓon the skew product𝐺 ×_𝜌Γgiven by 𝛾^′⋅ (𝑔, 𝛾) = (𝑔, 𝛾^′𝛾).

Proof. We check all the properties defining a groupoid action. First, we define the continuous open map

𝑝 ∶ 𝐺 ×_𝜌Γ → Γ⁽⁰⁾, 𝑝(𝑔, 𝛾) = 𝑡(𝛾)

and note that𝑑(𝛾^′) = 𝑡(𝛾) = 𝑝(𝑔, 𝛾). Now,𝑝(𝑔, 𝛾^′𝛾) = 𝑡(𝛾^′)and if(𝛾₁, 𝛾₂) ∈ Γ⁽²⁾, then

(𝛾₁𝛾₂) ⋅ (𝑔, 𝛾) = (𝑔, 𝛾₁𝛾₂𝛾) = 𝛾₁⋅ (𝛾₂⋅ (𝑔, 𝛾)).

Also, if(𝑔, 𝛾), (𝑔^′, 𝛾𝜌(𝑔)) ∈ 𝐺 ×_𝜌Γare composable with product(𝑔𝑔^′, 𝛾), then 𝛾^′⋅ (𝑔𝑔^′, 𝛾) = (𝑔𝑔^′, 𝛾^′𝛾) = (𝑔, 𝛾^′𝛾)(𝑔^′, 𝛾^′𝛾𝜌(𝑔)) = (𝛾^′⋅ (𝑔, 𝛾))(𝛾^′⋅ (𝑔^′, 𝛾𝜌(𝑔))).

The last condition to check is𝑡(𝛾) ⋅ (𝑔, 𝛾) = (𝑔, 𝛾), which is obvious.

We define a right action ofΓon𝐺 ×_𝜌Γby(𝑔, 𝛾) ⋅ 𝛾^′= (𝑔, 𝛾^′−1𝛾), and we form the semi-direct product(𝐺 ×_𝜌Γ) ⋊ Γmade of triples(𝑔, 𝛾, 𝛾^′) ∈ 𝐺 × Γ × Γsuch that(𝛾, 𝜌(𝑔)) ∈ Γ⁽²⁾ and𝑡(𝛾^′) = 𝑝(𝑔, 𝛾) = 𝑡(𝛾), with unit space𝐺⁽⁰⁾ ∗ Γand operations

(𝑔, 𝛾, 𝛾^′)(𝑔^′, 𝛾^′−1𝛾𝜌(𝑔), 𝛾^′′) = (𝑔𝑔^′, 𝛾, 𝛾^′𝛾^′′),

(𝑔, 𝛾, 𝛾^′)⁻¹= ((𝑔⁻¹, 𝛾𝜌(𝑔)) ⋅ 𝛾^′, 𝛾^′−1) = (𝑔⁻¹, 𝛾^′−1𝛾𝜌(𝑔), 𝛾^′−1).

For the next result, which resembles Takai duality, see Definition 1.3 in [17], Proposition 3.7 in [12] and Definition 3.1 in [9].

Theorem 3.8. Let𝐺, 𝐻, Γ be étale groupoids such that𝐺 acts on𝐻 and such that𝜌 ∶ 𝐺 → Γis a groupoid homomorphism. Then, using the above notation, (𝐻 ⋊ 𝐺) ×_𝜋𝐺is similar to𝐻and(𝐺 ×_𝜌Γ) ⋊ Γis similar to𝐺.

Proof. Recall that two (continuous) groupoid homomorphisms𝜙₁, 𝜙₂∶ 𝐺₁→ 𝐺₂are similar if there is a continuous function𝜃 ∶ 𝐺₁⁽⁰⁾→ 𝐺₂such that

𝜃(𝑡(𝑔))𝜙₁(𝑔) = 𝜙₂(𝑔)𝜃(𝑑(𝑔))

for all𝑔 ∈ 𝐺₁. Two topological groupoids𝐺₁, 𝐺₂are similar if there exist con- tinuous homomorphisms𝜙 ∶ 𝐺₁ → 𝐺₂ and𝜓 ∶ 𝐺₂ → 𝐺₁ such that 𝜓◦𝜙is similar to𝑖𝑑_𝐺1 and𝜙◦𝜓is similar to𝑖𝑑_𝐺2.

To show that(𝐻 ⋊ 𝐺) ×_𝜋𝐺is similar to𝐻, we define 𝜙 ∶ (𝐻 ⋊ 𝐺) ×_𝜋𝐺 → 𝐻, 𝜙(ℎ, 𝑔, 𝑔^′) = ℎ ⋅ 𝑔^′−1, 𝜓 ∶ 𝐻 → (𝐻 ⋊ 𝐺) ×_𝜋𝐺, 𝜓(ℎ) = (ℎ, 𝑝(ℎ), 𝑝(ℎ)) and

𝜃 ∶ 𝐻⁽⁰⁾ ∗ 𝐺 → (𝐻 ⋊ 𝐺) ×_𝜋𝐺, 𝜃(𝑢, 𝑔) = (𝑢 ⋅ 𝑔⁻¹, 𝑔, 𝑝(𝑢)).

We check that

(𝜙◦𝜓)(ℎ) = 𝜙(ℎ, 𝑝(ℎ), 𝑝(ℎ)) = ℎ ⋅ 𝑝(ℎ)⁻¹= ℎ and that

(∗) 𝜃[𝑡(ℎ, 𝑔, 𝑔^′)](ℎ, 𝑔, 𝑔^′) = (𝜓◦𝜙)(ℎ, 𝑔, 𝑔^′)𝜃[𝑑(ℎ, 𝑔, 𝑔^′)].

We have

𝑡(ℎ, 𝑔, 𝑔^′) = (ℎ, 𝑔, 𝑔^′)(ℎ, 𝑔, 𝑔^′)⁻¹= (ℎ, 𝑔, 𝑔^′)(ℎ⁻¹⋅ 𝑔, 𝑔⁻¹, 𝑔^′𝑔) =

= (ℎ(ℎ⁻¹⋅ 𝑔 ⋅ 𝑔⁻¹), 𝑔𝑔⁻¹, 𝑔^′) = (𝑡(ℎ), 𝑡(𝑔), 𝑔^′) ≡ (𝑡(ℎ), 𝑔^′), and

𝜃(𝑡(ℎ), 𝑔^′) = (𝑡(ℎ) ⋅ 𝑔^′−1, 𝑔^′, 𝑝(𝑡(ℎ)).

The left-hand side of(∗)becomes

(𝑡(ℎ) ⋅ 𝑔^′−1, 𝑔^′, 𝑝(𝑡(ℎ))(ℎ, 𝑔, 𝑔^′) = (ℎ ⋅ 𝑔^′−1, 𝑔^′𝑔, 𝑝(𝑡(ℎ)) = (ℎ ⋅ 𝑔^′−1, 𝑔^′𝑔, 𝑡(𝑔^′)).

Now,

(𝜓◦𝜙)(ℎ, 𝑔, 𝑔^′) = 𝜓(ℎ⋅𝑔^′−1) = (ℎ⋅𝑔^′−1, 𝑝(ℎ⋅𝑔^′−1), 𝑝(ℎ⋅𝑔^′−1) = (ℎ⋅𝑔^′−1, 𝑡(𝑔^′), 𝑡(𝑔^′)), 𝑑(ℎ, 𝑔, 𝑔^′) = (ℎ, 𝑔, 𝑔^′)⁻¹(ℎ, 𝑔, 𝑔^′) = (ℎ⁻¹⋅ 𝑔, 𝑔⁻¹, 𝑔^′𝑔)(ℎ, 𝑔, 𝑔^′) =

= ((ℎ⁻¹⋅ 𝑔)(ℎ ⋅ 𝑔), 𝑔⁻¹𝑔, 𝑔^′𝑔) = (𝑑(ℎ) ⋅ 𝑔, 𝑑(𝑔), 𝑔^′𝑔) ≡ (𝑑(ℎ) ⋅ 𝑔, 𝑔^′𝑔), and

𝜃(𝑑(ℎ) ⋅ 𝑔, 𝑔^′𝑔) = (𝑑(ℎ) ⋅ 𝑔 ⋅ (𝑔^′𝑔)⁻¹, 𝑔^′𝑔), 𝑝(𝑑(ℎ) ⋅ 𝑔)) = (𝑑(ℎ) ⋅ 𝑔^′−1, 𝑔^′𝑔), 𝑑(𝑔)).

The right-hand side becomes

(ℎ ⋅ 𝑔^′−1, 𝑡(𝑔^′), 𝑡(𝑔^′))(𝑑(ℎ) ⋅ 𝑔^′−1, 𝑔^′𝑔, 𝑑(𝑔)) = (ℎ ⋅ 𝑔^′−1, 𝑔^′𝑔, 𝑡(𝑔^′)), so(∗)is verified.

To show that(𝐺 ×_𝜌Γ) ⋊ Γis similar to𝐺, we define 𝜙 ∶ (𝐺 ×_𝜌Γ) ⋊ Γ → 𝐺, 𝜙(𝑔, 𝛾, 𝛾^′) = 𝑔, 𝜓 ∶ 𝐺 → (𝐺 ×_𝜌Γ) ⋊ Γ, 𝜓(𝑔) = (𝑔, 𝜌(𝑡(𝑔)), 𝜌(𝑔)) and

𝜃 ∶ 𝐺⁽⁰⁾ ∗ Γ → (𝐺 ×_𝜌Γ) ⋊ Γ, 𝜃(𝑢, 𝛾) = (𝑢, 𝜌(𝑢), 𝛾⁻¹).

We have

(𝜙◦𝜓)(𝑔) = 𝜙(𝑔, 𝜌(𝑡(𝑔)), 𝜌(𝑔)) = 𝑔 and we need to verify that

(∗∗) 𝜃(𝑡(𝑔, 𝛾, 𝛾^′))(𝑔, 𝛾, 𝛾^′) = (𝜓◦𝜙)(𝑔, 𝛾, 𝛾^′)𝜃(𝑑(𝑔, 𝛾, 𝛾^′)).

We compute

𝑡(𝑔, 𝛾, 𝛾^′) = (𝑔, 𝛾, 𝛾^′)(𝑔, 𝛾, 𝛾^′)⁻¹= (𝑔, 𝛾, 𝛾^′)(𝑔⁻¹, 𝛾^′−1𝛾𝜌(𝑔), 𝛾^′−1) =

= (𝑡(𝑔), 𝛾, 𝑡(𝛾^′)) ≡ (𝑡(𝑔), 𝛾) and

𝜃(𝑡(𝑔), 𝛾) = (𝑡(𝑔), 𝜌(𝑡(𝑔), 𝛾⁻¹), so the left-hand side of(∗∗)becomes

(𝑡(𝑔), 𝜌(𝑡(𝑔)), 𝛾⁻¹)(𝑔, 𝛾, 𝛾^′) = (𝑔, 𝜌(𝑡(𝑔)), 𝛾⁻¹𝛾^′).

Also

(𝜓◦𝜙)(𝑔, 𝛾, 𝛾^′) = (𝑔, 𝜌(𝑡(𝑔)), 𝜌(𝑔)),

𝑑(𝑔, 𝛾, 𝛾^′) = (𝑔, 𝛾, 𝛾^′)⁻¹(𝑔, 𝛾, 𝛾^′) = (𝑔⁻¹, 𝛾^′−1𝛾𝜌(𝑔), 𝛾^′−1)(𝑔, 𝛾, 𝛾^′) =

= (𝑑(𝑔), 𝛾^′−1𝛾𝜌(𝑔), 𝑑(𝛾^′)) ≡ (𝑑(𝑔), 𝛾^′−1𝛾𝜌(𝑔)), 𝜃(𝑑(𝑔), 𝛾^′−1𝛾𝜌(𝑔)) = (𝑑(𝑔), 𝜌(𝑑(𝑔), 𝜌(𝑔)⁻¹𝛾⁻¹𝛾^′), and the right-hand side is

(𝑔, 𝜌(𝑡(𝑔)), 𝜌(𝑔))(𝑑(𝑔), 𝜌(𝑑(𝑔)), 𝜌(𝑔)⁻¹𝛾⁻¹𝛾^′) = (𝑔, 𝜌(𝑡(𝑔)), 𝛾⁻¹𝛾^′),

so(∗∗)is verified.

For skew products and semi-direct products of groupoids, there are Lyndon–

Hochschild–Serre spectral sequences for the computation of their homology, see [3] and [12].

Theorem 3.9. Let𝐺, Γbe étale groupoids.

(1) Suppose that𝜌 ∶ 𝐺 → Γis a groupoid homomorphism. Then there exists a spectral sequence

𝐸²_𝑝,𝑞= 𝐻_𝑝(Γ, 𝐻_𝑞(𝐺 ×_𝜌Γ)) ⇒ 𝐻_𝑝+𝑞(𝐺),

where𝐻_𝑞(𝐺 ×_𝜌Γ)is regarded as aΓ-module via the action𝜌 ∶ Γ ↷ 𝐺 ×̂ _𝜌Γ.

(2) Suppose that𝜑 ∶ Γ ↷ 𝐺is a groupoid action. Then there exists a spectral sequence

𝐸_𝑝,𝑞² = 𝐻_𝑝(Γ, 𝐻_𝑞(𝐺)) ⇒ 𝐻_𝑝+𝑞(Γ ⋉ 𝐺), where𝐻_𝑞(𝐺)is regarded as aΓ-module via the action𝜑.

Note that for𝐺an ample Hausdorff groupoid and𝜌 ∶ 𝐺 → ℤa cocycle, we have the following long exact sequence involving the homology of𝐺 and the homology of𝐺 ×_𝜌ℤ,

0 ⟵ 𝐻₀(𝐺) ⟵ 𝐻₀(𝐺 ×_𝜌ℤ)^{𝑖𝑑−𝜌}⟵ 𝐻^∗ ₀(𝐺 ×_𝜌ℤ) ⟵ 𝐻₁(𝐺) ⟵ ⋯

⋯ ⟵ 𝐻_𝑛(𝐺) ⟵ 𝐻_𝑛(𝐺 ×_𝜌ℤ)^{𝑖𝑑−𝜌}⟵ 𝐻^∗ _𝑛(𝐺 ×_𝜌ℤ) ⟵ 𝐻_𝑛+1(𝐺) ⟵ ⋯ Here𝜌_∗is the map induced by the action𝜌 ∶ ℤ ↷ 𝐺 ×̂ _𝜌ℤ(see Lemma 1.3 in [15]).

4. Self-similar groupoid actions and their𝑪^∗-algebra

We recall some facts about self-similar groupoid actions and their Cuntz- Pimsner algebras from [10]. Let 𝐸 = (𝐸⁰, 𝐸¹, 𝑟, 𝑠) be a finite directed graph with no sources. For𝑘 ≥ 2, define the set of paths of length𝑘in𝐸as

𝐸^𝑘 = {𝑒₁𝑒₂⋯ 𝑒_𝑘 ∶ 𝑒_𝑖 ∈ 𝐸¹, 𝑟(𝑒_𝑖+1) = 𝑠(𝑒_𝑖)}.

The maps𝑟, 𝑠are naturally extended to𝐸^𝑘by taking

𝑟(𝑒₁𝑒₂⋯ 𝑒_𝑘) = 𝑟(𝑒₁), 𝑠(𝑒₁𝑒₂⋯ 𝑒_𝑘) = 𝑠(𝑒_𝑘).

We denote by𝐸^∗ ∶=⋃

𝑘≥0𝐸^𝑘 the space of finite paths (including vertices) and by𝐸^∞the infinite path space of𝐸with the usual topology given by the cylinder sets𝑍(𝛼) = {𝛼𝜉 ∶ 𝜉 ∈ 𝐸^∞}for𝛼 ∈ 𝐸^∗.

We can visualize the set𝐸^∗as indexing the vertices of a union of rooted trees or forest𝑇_𝐸 given by𝑇_𝐸⁰ = 𝐸^∗and with edges

𝑇¹_𝐸 = {(𝜇, 𝜇𝑒) ∶ 𝜇 ∈ 𝐸^∗, 𝑒 ∈ 𝐸¹and𝑠(𝜇) = 𝑟(𝑒)}.

Example4.1. For the graph

𝑢 𝑣 𝑤

𝑒2

𝑒₃

𝑒4

𝑒₅ 𝑒₆ 𝑒₁

the forest𝑇_𝐸 looks like

𝑢 𝑣 𝑤

𝑒1 𝑒3

𝑒₁𝑒₁

⋮

𝑒₁𝑒₃

⋮

𝑒₃𝑒₂

⋮

𝑒₃𝑒₆

⋮

𝑒2 𝑒6

𝑒₂𝑒₁

⋮

𝑒₂𝑒₃

⋮

𝑒₆𝑒₄

⋮

𝑒₆𝑒₅

⋮

𝑒4 𝑒5

𝑒₄𝑒₂

⋮

𝑒₄𝑒₆

⋮

𝑒₅𝑒₂

⋮

𝑒₅𝑒₆

⋮

Recall that a partial isomorphism of the forest𝑇_𝐸 corresponding to a given directed graph𝐸consists of a pair(𝑣, 𝑤) ∈ 𝐸⁰× 𝐸⁰and a bijection𝑔 ∶ 𝑣𝐸^∗ → 𝑤𝐸^∗such that

∙ 𝑔|_𝑣𝐸𝑘 ∶ 𝑣𝐸^𝑘 → 𝑤𝐸^𝑘 is bijective for all𝑘 ≥ 1.

∙ 𝑔(𝜇𝑒) ∈ 𝑔(𝜇)𝐸¹for𝜇 ∈ 𝑣𝐸^∗and𝑒 ∈ 𝐸¹with𝑟(𝑒) = 𝑠(𝜇).

The set of partial isomorphisms of𝑇_𝐸 forms a groupoid PIso(𝑇_𝐸)with unit space𝐸⁰. The identity morphisms are𝑖𝑑_𝑣 ∶ 𝑣𝐸^∗ → 𝑣𝐸^∗, the inverse of𝑔 ∶ 𝑣𝐸^∗ → 𝑤𝐸^∗is𝑔⁻¹ ∶ 𝑤𝐸^∗ → 𝑣𝐸^∗, and the multiplication is composition. We often identify𝑣 ∈ 𝐸⁰with𝑖𝑑_𝑣∈PIso(𝑇_𝐸).

Definition 4.2. Let𝐸be a finite directed graph with no sources, and let𝐺be a groupoid with unit space𝐸⁰. Aself-similar action(𝐺, 𝐸)on the path space of 𝐸is given by a faithful groupoid homomorphism𝐺 →PIso(𝑇_𝐸)such that for every𝑔 ∈ 𝐺and every𝑒 ∈ 𝑑(𝑔)𝐸¹there exists a uniqueℎ ∈ 𝐺denoted by𝑔|_𝑒 and called the restriction of𝑔to𝑒such that

𝑔 ⋅ (𝑒𝜇) = (𝑔 ⋅ 𝑒)(ℎ ⋅ 𝜇) for all 𝜇 ∈ 𝑠(𝑒)𝐸^∗.

Remark4.3. It is possible that𝑔|_𝑒 = 𝑔for all𝑒 ∈ 𝑑(𝑔)𝐸¹, in which case 𝑔 ⋅ (𝑒₁𝑒₂⋯ 𝑒_𝑛) = (𝑔 ⋅ 𝑒₁) ⋯ (𝑔 ⋅ 𝑒_𝑛).

We have

𝑑(𝑔|_𝑒) = 𝑠(𝑒), 𝑡(𝑔|_𝑒) = 𝑠(𝑔 ⋅ 𝑒) = 𝑔|_𝑒⋅ 𝑠(𝑒), 𝑟(𝑔 ⋅ 𝑒) = 𝑔 ⋅ 𝑟(𝑒).

In particular, the source map may not be equivariant as in [7]. It is shown in Appendix A of [10] that a self-similar group action(𝐺, 𝐸)as in [7] determines a self-similar groupoid action(𝐸⁰⋊ 𝐺, 𝐸) as in Definition4.2, where𝐸⁰⋊ 𝐺 is the semi-direct product or the action groupoid of the group𝐺acting on𝐸⁰. Note that not any self-similar groupoid action comes from a self-similar group action, as seen in our example below.

Proposition 4.4. A self-similar groupoid action (𝐺, 𝐸)as above extends to an action of𝐺on the path space𝐸^∗and determines an action of𝐺on the graph𝑇_𝐸, in the sense that𝐺 acts on both the vertex space𝑇_𝐸⁰ and the edge space𝑇¹_𝐸 and intertwines the range and the source maps of𝑇_𝐸, see Definition 4.1 in[5].

Proof. Indeed, the vertex space𝑇⁰_𝐸 = 𝐸^∗is fibered over𝐺⁽⁰⁾ = 𝐸⁰via the map 𝜇 ↦ 𝑟(𝜇). For (𝜇, 𝜇𝑒) ∈ 𝑇_𝐸¹ we set 𝑠(𝜇, 𝜇𝑒) = 𝜇𝑒 and 𝑟(𝜇, 𝜇𝑒) = 𝜇. Since 𝑟(𝜇𝑒) = 𝑟(𝜇), the edge space𝑇_𝐸¹ is also fibered over𝐺⁽⁰⁾. The action of𝐺on𝑇_𝐸¹ is given by

𝑔 ⋅ (𝜇, 𝜇𝑒) = (𝑔 ⋅ 𝜇, 𝑔 ⋅ (𝜇𝑒)) when 𝑑(𝑔) = 𝑟(𝜇).

Since

𝑠(𝑔 ⋅ (𝜇, 𝜇𝑒)) = 𝑠(𝑔 ⋅ 𝜇, 𝑔 ⋅ (𝜇𝑒)) = 𝑔 ⋅ (𝜇𝑒) = 𝑔 ⋅ 𝑠(𝜇, 𝜇𝑒) and

𝑟(𝑔 ⋅ (𝜇, 𝜇𝑒)) = 𝑟(𝑔 ⋅ 𝜇, 𝑔 ⋅ (𝜇𝑒)) = 𝑔 ⋅ 𝜇 = 𝑔 ⋅ 𝑟(𝜇, 𝜇𝑒),

the actions on𝑇⁰_𝐸and𝑇_𝐸¹ are compatible.

The faithfulness condition ensures that for each𝑔 ∈ 𝐺and each𝜇 ∈ 𝐸^∗with 𝑑(𝑔) = 𝑟(𝜇), there is a unique element𝑔|_𝜇 ∈ 𝐺satisfying

𝑔 ⋅ (𝜇𝜈) = (𝑔 ⋅ 𝜇)(𝑔|_𝜇⋅ 𝜈) for all𝜈 ∈ 𝑠(𝜇)𝐸^∗.

By Proposition 3.6 of [10], self-similar groupoid actions have the following prop- erties: for𝑔, ℎ ∈ 𝐺, 𝜇 ∈ 𝑑(𝑔)𝐸^∗, and𝜈 ∈ 𝑠(𝜇)𝐸^∗,

(1)𝑔|_𝜇𝜈 = (𝑔|_𝜇)|_𝜈; (2)𝑖𝑑_𝑟(𝜇)|_𝜇 = 𝑖𝑑_𝑠(𝜇);

(3) if(ℎ, 𝑔) ∈ 𝐺⁽²⁾, then(ℎ|_𝑔⋅𝜇, 𝑔|_𝜇) ∈ 𝐺⁽²⁾and(ℎ𝑔)|_𝜇 = (ℎ|_𝑔⋅𝜇)(𝑔|_𝜇); (4)𝑔⁻¹|_𝜇 = (𝑔|_𝑔−1⋅𝜇)⁻¹.

Definition 4.5. The𝐶^∗-algebra𝐶^∗(𝐺, 𝐸)of a self-similar action(𝐺, 𝐸)is de- fined as the Cuntz-Pimsner algebra of the𝐶^∗-correspondence

ℳ = ℳ(𝐺, 𝐸) = 𝒳(𝐸) ⊗_𝐶(𝐸0)𝐶^∗(𝐺)

over𝐶^∗(𝐺). Here𝒳(𝐸) = 𝐶(𝐸¹)is the𝐶^∗-correspondence over𝐶(𝐸⁰)associ- ated to the graph𝐸and𝐶(𝐸⁰) = 𝐶(𝐺⁽⁰⁾) ⊆ 𝐶^∗(𝐺). The right action of𝐶^∗(𝐺)on ℳis the usual one and the left action is determined by the representation

𝑊 ∶ 𝐺 → ℒ(ℳ), 𝑊_𝑔(𝑖_𝑒⊗ 𝑎) = {𝑖_𝑔⋅𝑒⊗ 𝑖_𝑔|

𝑒𝑎 if𝑑(𝑔) = 𝑟(𝑒) 0 otherwise,

where𝑔 ∈ 𝐺, 𝑖_𝑒 ∈ 𝐶(𝐸¹)and𝑖_𝑔 ∈ 𝐶_𝑐(𝐺)are point masses and𝑎 ∈ 𝐶^∗(𝐺). The inner product ofℳis given by

⟨𝜉 ⊗ 𝑎, 𝜂 ⊗ 𝑏⟩ = ⟨⟨𝜂, 𝜉⟩𝑎, 𝑏⟩ = 𝑎^∗⟨𝜉, 𝜂⟩𝑏 for𝜉, 𝜂 ∈ 𝐶(𝐸¹)and𝑎, 𝑏 ∈ 𝐶^∗(𝐺).

Remark4.6. Recall that the operations on𝒳(𝐸)are given by (𝜉 ⋅ 𝑎)(𝑒) = 𝜉(𝑒)𝑎(𝑠(𝑒)), ⟨𝜉, 𝜂⟩(𝑣) = ∑

𝑠(𝑒)=𝑣

𝜉(𝑒)𝜂(𝑒), (𝑎 ⋅ 𝜉)(𝑒) = 𝑎(𝑟(𝑒))𝜉(𝑒)

for𝑎 ∈ 𝐶(𝐸⁰)and𝜉, 𝜂 ∈ 𝐶(𝐸¹). The elements𝑖_𝑒⊗ 1for𝑒 ∈ 𝐸¹form a Parseval frame forℳand every𝜁 ∈ ℳis a finite sum

𝜁 = ∑

𝑒∈𝐸¹

𝑖_𝑒⊗ ⟨𝑖_𝑒⊗ 1, 𝜁⟩.

In particular, if𝒳(𝐸)^∗denotes the dual𝐶^∗-correspondence, then

ℒ(ℳ) = 𝒦(ℳ) ≅ 𝒳(𝐸) ⊗_𝐶(𝐸⁰₎𝐶^∗(𝐺) ⊗_𝐶(𝐸⁰₎𝒳(𝐸)^∗ ≅ 𝑀_𝑛⊗ 𝐶^∗(𝐺), where𝑛 = |𝐸¹|. The isomorphism is given by

𝑖_𝑒𝑗⊗ 𝑖_𝑔⊗ 𝑖_𝑒^∗_𝑘 ↦ 𝑒_𝑗𝑘⊗ 𝑖_𝑔

for𝐸¹= {𝑒₁, ..., 𝑒_𝑛}and for matrix units𝑒_𝑗𝑘 ∈ 𝑀_𝑛. There is a unital homomor- phism𝒦(𝒳(𝐸)) → 𝒦(ℳ)given by

𝑖_𝑒⊗ 𝑖^∗_𝑓→ 𝑖_𝑒⊗ 1 ⊗ 𝑖_𝑓^∗.

Since our groupoids have finite unit space𝐸⁰, the orbit space for the canon- ical action of 𝐺 on 𝐸⁰ is finite, and 𝐶^∗(𝐺) is the direct sum of 𝐶^∗-algebras of transitive groupoids. Each such transitive groupoid will be isomorphic to a groupoid of the form𝑉 × 𝐻 × 𝑉 with the usual operations, for some sub- set𝑉 ⊆ 𝐸⁰ and isotropy group𝐻, hence its𝐶^∗-algebra will be isomorphic to 𝐶^∗(𝐻) ⊗ 𝑀_|𝑉|.

We recall the following result, see Propositions 4.4 and 4.7 in [10].

Theorem 4.7. If𝑈_𝑔, 𝑃_𝑣and𝑇_𝑒 are the images of𝑔 ∈ 𝐺, 𝑣 ∈ 𝐸⁰ = 𝐺⁽⁰⁾and of 𝑒 ∈ 𝐸¹in the Cuntz-Pimsner algebra𝐶^∗(𝐺, 𝐸), then

∙ 𝑔 ↦ 𝑈_𝑔is a representation by partial isometries of𝐺with𝑈_𝑣 = 𝑃_𝑣 for 𝑣 ∈ 𝐸⁰;

∙ 𝑇_𝑒are partial isometries with𝑇_𝑒^∗𝑇_𝑒= 𝑃_𝑠(𝑒)and ∑

𝑟(𝑒)=𝑣

𝑇_𝑒𝑇^∗_𝑒 = 𝑃_𝑣;

∙ 𝑈_𝑔𝑇_𝑒 = {𝑇_𝑔⋅𝑒𝑈_𝑔|𝑒 if𝑑(𝑔) = 𝑟(𝑒)

0, otherwise and 𝑈_𝑔𝑃_𝑣 = {𝑃_𝑔⋅𝑣𝑈_𝑔 if𝑑(𝑔) = 𝑣 0, otherwise.

There is a gauge action𝛾of𝕋on𝐶^∗(𝐺, 𝐸)such that𝛾_𝑧(𝑈_𝑔) = 𝑈_𝑔,and𝛾_𝑧(𝑇_𝑒) = 𝑧𝑇_𝑒for𝑧 ∈ 𝕋.

Given𝜇 = 𝑒₁⋯ 𝑒_𝑛 ∈ 𝐸^∗with𝑒_𝑖 ∈ 𝐸¹, we let𝑇_𝜇 ∶= 𝑇_𝑒1⋯ 𝑇_𝑒𝑛. Then𝐶^∗(𝐺, 𝐸) is the closed linear span of elements𝑇_𝜇𝑈_𝑔𝑇^∗_𝜈, where𝜇, 𝜈 ∈ 𝐸^∗and𝑔 ∈ 𝐺_𝑠(𝜈)^𝑠(𝜇).

For each𝑘 ≥ 1, considerℱ_𝑘the closed linear span of elements𝑇_𝜇𝑈_𝑔𝑇_𝜈^∗with 𝜇, 𝜈 ∈ 𝐸^𝑘 and𝑔 ∈ 𝐺_𝑠(𝜈)^𝑠(𝜇). Then the fixed point algebraℱ(𝐺, 𝐸) ∶= 𝐶^∗(𝐺, 𝐸)^𝕋 under the gauge action is isomorphic tolim

,,→ℱ_𝑘. We have

ℱ_𝑘 ≅ ℒ(ℳ^⊗𝑘) ≅ 𝒳(𝐸)^⊗𝑘⊗_𝐶(𝐸⁰₎𝐶^∗(𝐺) ⊗_𝐶(𝐸⁰₎𝒳(𝐸)^∗⊗𝑘

using the map𝑇_𝜇𝑈_𝑔𝑇_𝜈^∗ ↦ 𝑖_𝜇⊗ 𝑖_𝑔⊗ 𝑖^∗_𝜈, where𝑖_𝜇 ∈ 𝒳(𝐸)^⊗𝑘 = 𝐶(𝐸^𝑘)are point masses. The embeddingsℱ_𝑘 ↪ ℱ_𝑘+1are determined by the map

𝜙 = 𝜙_𝑊 ∶ 𝐶^∗(𝐺) → ℒ(ℳ), 𝜙_𝑊(𝑖_𝑔) = 𝑊_𝑔. In particular, for𝑎 ∈ 𝐶^∗(𝐺)we get

𝜙(𝑎) = ∑

𝑒∈𝐸¹

𝜃_{𝑖𝑒⊗1,𝑎}^∗_{(𝑖𝑒⊗1)},

where𝜃_𝜉,𝜂(𝜁) = 𝜉⟨𝜂, 𝜁⟩. The embeddingsℱ_𝑘 ↪ ℱ_𝑘+1are then

𝜙_𝑘(𝑖_𝜇⊗ 𝑖_𝑔⊗ 𝑖_𝜈^∗) =

⎧

⎨

⎩

∑

𝑥∈𝑑(𝑔)𝐸¹

𝑖_𝜇𝑦⊗ 𝑖_𝑔|

𝑥 ⊗ 𝑖^∗_𝜈𝑥, if𝑔 ∈ 𝐺^𝑠(𝜇)

𝑠(𝜈) and𝑔 ⋅ 𝑥 = 𝑦 0, otherwise.

Remark4.8. The𝐶^∗-algebra𝐶^∗(𝐺, 𝐸)can be described as the crossed product ofℱ(𝐺, 𝐸)by an endomorphism and in many cases, knowledge about𝐾_∗(ℱ_𝑘) is sufficient to determine𝐾_∗(ℱ(𝐺, 𝐸))and𝐾_∗(𝐶^∗(𝐺, 𝐸)). For the case when𝐺 is a group, see section 3 in [14]. In the particular case when𝑔|_𝑒= 𝑔for all𝑔 ∈ 𝐺 and𝑒 ∈ 𝑑(𝑔)𝐸¹we have𝐶^∗(𝐺, 𝐸) ≅ 𝐶^∗(𝐸) ⋊ 𝐺.

5. Exel-Pardo groupoids for self-similar actions

In this section, we generalize results from [7] and we define the groupoid associated to a self-similar action of a groupoid𝐺on the path space of a finite directed graph𝐸with no sources.

As in [7], we first define the inverse semigroup

𝒮(𝐺, 𝐸) = {(𝛼, 𝑔, 𝛽) ∶ 𝛼, 𝛽 ∈ 𝐸^∗, 𝑔 ∈ 𝐺_𝑠(𝛽)^𝑠(𝛼)} ∪ {0}

associated to the self-similar action(𝐺, 𝐸), with operations

(𝛼, 𝑔, 𝛽)(𝜆, ℎ, 𝜔) =

⎧

⎨

⎩

(𝛼, 𝑔(ℎ|_ℎ⁻¹_⋅𝜇), 𝜔(ℎ⁻¹⋅ 𝜇)) if𝛽 = 𝜆𝜇 (𝛼(𝑔 ⋅ 𝜇), 𝑔|_𝜇ℎ, 𝜔) if𝜆 = 𝛽𝜇

0 otherwise

and(𝛼, 𝑔, 𝛽)^∗ = (𝛽, 𝑔⁻¹, 𝛼)for𝛼, 𝛽, 𝜆, 𝜔 ∈ 𝐸^∗. These operations make sense since

𝑑(𝑔) = 𝑠(𝛽) = 𝑡(ℎ|_ℎ⁻¹_⋅𝜇)and𝑑(𝑔(ℎ|_ℎ⁻¹_⋅𝜇)) = 𝑠(𝜔(ℎ⁻¹⋅ 𝜇))when𝛽 = 𝜆𝜇, 𝑑(𝑔|_𝜇) = 𝑠(𝜇) = 𝑡(ℎ)and𝑑(𝑔|_𝜇ℎ) = 𝑠(𝜔) when𝜆 = 𝛽𝜇.

Note that(𝛼, 𝑔, 𝛽)(𝛽, ℎ, 𝜔) = (𝛼, 𝑔ℎ, 𝜔)and the nonzero idempotents are of the form𝑧_𝛼= (𝛼, 𝑠(𝛼), 𝛼).

The inverse semigroup𝒮(𝐺, 𝐸)acts on the infinite path space𝐸^∞by partial homeomorphisms. The action of(𝛼, 𝑔, 𝛽) ∈ 𝒮(𝐺, 𝐸)on𝜉 = 𝛽𝜇 ∈ 𝛽𝐸^∞is given by

(𝛼, 𝑔, 𝛽) ⋅ 𝛽𝜇 = 𝛼(𝑔 ⋅ 𝜇) ∈ 𝛼𝐸^∞.

The action of𝐺on𝐸^∞is defined by𝑔 ⋅ 𝜇 = 𝜂, where for all𝑛we have𝜂₁⋯ 𝜂_𝑛= 𝑔 ⋅ (𝜇₁⋯ 𝜇_𝑛). Note that𝑟(𝑔 ⋅ 𝜇) = 𝑔 ⋅ 𝑟(𝜇) = 𝑔 ⋅ 𝑠(𝛽) = 𝑠(𝛼), so the action is well defined.

The groupoid of germs associated with(𝒮(𝐺, 𝐸), 𝐸^∞)is

𝒢(𝐺, 𝐸) = {[𝛼, 𝑔, 𝛽; 𝜉] ∶ 𝛼, 𝛽 ∈ 𝐸^∗, 𝑔 ∈ 𝐺_𝑠(𝛽)^𝑠(𝛼), 𝜉 ∈ 𝛽𝐸^∞}.

Two germs[𝛼, 𝑔, 𝛽; 𝜉], [𝛼^′, 𝑔^′, 𝛽^′; 𝜉^′]in𝒢(𝐺, 𝐸)are equal if and only if𝜉 = 𝜉^′ and there exists a neighborhood𝑉of𝜉such that(𝛼, 𝑔, 𝛽) ⋅ 𝜂 = (𝛼^′, 𝑔^′, 𝛽^′) ⋅ 𝜂for all𝜂 ∈ 𝑉. We obtain that𝜉 = 𝛽𝜆𝜁 for𝜆 ∈ 𝐸^∗ and𝜁 ∈ 𝐸^∞, with𝑟(𝜆) = 𝑠(𝛽) and𝑟(𝜁) = 𝑠(𝜆). Moreover,

𝛼^′ = 𝛼(𝑔 ⋅ 𝜆), 𝛽^′= 𝛽𝜆, and𝑔^′= 𝑔|_𝜆. The unit space of𝒢(𝐺, 𝐸)is

𝒢(𝐺, 𝐸)⁽⁰⁾ = {[𝛼, 𝑠(𝛼), 𝛼; 𝜉] ∶ 𝜉 ∈ 𝛼𝐸^∞}, identified with𝐸^∞by the map[𝛼, 𝑠(𝛼), 𝛼; 𝜉] ↦ 𝜉.

The terminus and domain maps of the groupoid𝒢(𝐺, 𝐸)are given by 𝑡([𝛼, 𝑔, 𝛽; 𝛽𝜇]) = 𝛼(𝑔 ⋅ 𝜇), 𝑑([𝛼, 𝑔, 𝛽; 𝛽𝜇]) = 𝛽𝜇.

If two elements𝛾₁, 𝛾₂ ∈ 𝒢(𝐺, 𝐸)are composable, then

𝛾₁ = [𝛼₁, 𝑔₁, 𝛼₂; 𝛼₂(𝑔₂⋅ 𝜉)], 𝛾₂= [𝛼₂, 𝑔₂, 𝛽; 𝛽𝜉]

for some𝛼₁, 𝛼₂, 𝛽 ∈ 𝐸^∗, 𝜉 ∈ 𝐸^∞, (𝑔₁, 𝑔₂) ∈ 𝐺⁽²⁾and in this case 𝛾₁𝛾₂= [𝛼₁, 𝑔₁𝑔₂, 𝛽; 𝛽𝜉].

In particular,

[𝛼, 𝑔, 𝛽; 𝛽𝜇]⁻¹= [𝛽, 𝑔⁻¹, 𝛼; 𝛼(𝑔 ⋅ 𝜇)].

The topology on𝒢(𝐺, 𝐸)is generated by the compact open bisections of the form

𝐵(𝛼, 𝑔, 𝛽; 𝑉) = {[𝛼, 𝑔, 𝛽; 𝜉] ∈ 𝒢(𝐺, 𝐸) ∶ 𝜉 = 𝛽𝜁 ∈ 𝑉},

where𝛼, 𝛽 ∈ 𝐸^∗, 𝑔 ∈ 𝐺_𝑠(𝛽)^𝑠(𝛼)are fixed, and𝑉 ⊆ 𝑍(𝛽) = 𝛽𝐸^∞is an open subset.

Definition 5.1. A self-similar groupoid action(𝐺, 𝐸)is called pseudo free if for every𝑔 ∈ 𝐺and every𝑒 ∈ 𝑑(𝑔)𝐸¹, the condition𝑔 ⋅ 𝑒 = 𝑒and𝑔|_𝑒 = 𝑠(𝑒)implies that𝑔 = 𝑟(𝑒).

Remark5.2. If(𝐺, 𝐸)is pseudo free, then𝑔₁⋅ 𝛼 = 𝑔₂⋅ 𝛼and𝑔₁|_𝛼 = 𝑔₂|_𝛼 for some𝛼 ∈ 𝐸^∗implies𝑔₁= 𝑔₂.

Proof. Indeed, since𝑔⁻¹₂ 𝑔₁ ⋅ 𝛼 = 𝛼and𝑔⁻¹₂ 𝑔₁|_𝛼 = 𝑔⁻¹₂ |_𝑔1⋅𝛼𝑔₁|_𝛼 = 𝑑(𝑔₁|_𝛼) = 𝑠(𝛼), it follows that𝑔₂⁻¹𝑔₁= 𝑟(𝛼), so𝑔₁ = 𝑔₂. Theorem 5.3. If the action of𝐺on𝐸 is pseudo free, then the groupoid𝒢(𝐺, 𝐸) is Hausdorff and its𝐶^∗-algebra𝐶^∗(𝒢(𝐺, 𝐸))is isomorphic to the Cuntz-Pimsner algebra𝐶^∗(𝐺, 𝐸).

Proof. Since(𝐺, 𝐸) is pseudo free, it follows that[𝛼, 𝑔, 𝛽; 𝜉] = [𝛼, 𝑔^′, 𝛽; 𝜉]if and only if𝑔 = 𝑔^′. Moreover, the groupoid𝒢(𝐺, 𝐸)is Hausdorff, see Proposi- tion 12.1 in [7]. Using the properties given in Theorem 4.7and the groupoid multiplication, the isomorphism𝜙 ∶ 𝐶^∗(𝐺, 𝐸) → 𝐶^∗(𝒢(𝐺, 𝐸))is given by

𝜙(𝑃_𝑣) = 𝜒𝐵(𝑣,𝑣,𝑣;𝑍(𝑣)), 𝜙(𝑇_𝑒) = 𝜒𝐵(𝑒,𝑠(𝑒),𝑠(𝑒);𝑍(𝑠(𝑒))), 𝜙(𝑈_𝑔) = 𝜒𝐵(𝑡(𝑔),𝑔,𝑑(𝑔);𝑍(𝑑(𝑔)))

for𝑣 ∈ 𝐸⁰, 𝑒 ∈ 𝐸¹and𝑔 ∈ 𝐺. Here𝜒_𝐴is the indicator function of𝐴. Recall that ample Hausdorff groupoids which are similar or Morita equiva- lent have isomorphic homology, see Lemma 4.3 and Theorem 3.12 in [9]. The general strategy of computing the homology of the ample groupoid𝒢(𝐺, 𝐸)is the following.

There is a cocycle𝜌 ∶ 𝒢(𝐺, 𝐸) → ℤgiven by[𝛼, 𝑔, 𝛽; 𝜉] ↦ |𝛼| − |𝛽|with kernel

ℋ(𝐺, 𝐸) = {[𝛼, 𝑔, 𝛽; 𝜉] ∈ 𝒢(𝐺, 𝐸) ∶ |𝛼| = |𝛽|}.

It follows from Theorem3.9that we have a spectral sequence 𝐸_𝑝,𝑞² = 𝐻_𝑝(ℤ, 𝐻_𝑞(ℋ(𝐺, 𝐸))) ⇒ 𝐻_𝑝+𝑞(𝒢(𝐺, 𝐸)).

Nowℋ(𝐺, 𝐸) =⋃

𝑘≥1ℋ_𝑘(𝐺, 𝐸)where

ℋ_𝑘(𝐺, 𝐸) = {[𝛼, 𝑔, 𝛽; 𝜉] ∈ 𝒢(𝐺, 𝐸) ∶ |𝛼| = |𝛽| = 𝑘}.

There are groupoid homomorphisms

𝜏_𝑘 ∶ ℋ_𝑘(𝐺, 𝐸) → 𝐺, 𝜏_𝑘([𝛼, 𝑔, 𝛽; 𝜉]) = 𝑔

andker 𝜏_𝑘 is AF for all𝑘 ≥ 1. Indeed, consider𝑅_𝑘 the equivalence relation on𝐸^𝑘 such that(𝛼, 𝛽) ∈ 𝑅_𝑘 if there is𝑔 ∈ 𝐺with𝑔 ⋅ 𝑠(𝛽) = 𝑠(𝛼). Then the map[𝛼, 𝑔, 𝛽; 𝜉] ↦ ((𝜉, 𝑔), (𝛼, 𝛽))gives an isomorphism betweenℋ_𝑘(𝐺, 𝐸)and (𝐸^∞ ⋊ 𝐺) × 𝑅_𝑘, soker 𝜏_𝑘 is isomorphic to𝐸^∞ × 𝑅_𝑘. It follows that we have another spectral sequence

𝐸_𝑝,𝑞² = 𝐻_𝑝(𝐺, 𝐻_𝑞(ker 𝜏_𝑘)) ⇒ 𝐻_𝑝+𝑞(ℋ_𝑘(𝐺, 𝐸)).

It is known that 𝐻₀(ker 𝜏_𝑘) ≅ 𝐾₀(𝐶^∗(ker 𝜏_𝑘)) and 𝐻_𝑞(ker 𝜏_𝑘) = 0 for𝑘 ≥ 1. Also, ker 𝜏_𝑘 is similar with ℋ_𝑘(𝐺, 𝐸) ×_𝜏𝑘 𝐺. Assuming that we computed 𝐻_𝑞(ℋ_𝑘(𝐺, 𝐸))for all𝑘, then

𝐻_𝑞(ℋ(𝐺, 𝐸)) = lim

𝑘→∞,,→

𝐻_𝑞(ℋ_𝑘(𝐺, 𝐸)) can be computed using the inclusion maps

𝑗_𝑘 ∶ ℋ_𝑘(𝐺, 𝐸) ↪ ℋ_𝑘+1(𝐺, 𝐸), 𝑗_𝑘([𝛼, 𝑔, 𝛽; 𝛽𝑥𝜇]) = [𝛼𝑦, 𝑔|_𝑥, 𝛽𝑥; 𝛽𝑥𝜇], where𝑥 ∈ 𝐸¹and𝑔 ⋅ 𝑥 = 𝑦.

6. Example

Consider again the graph from Example4.1

𝑢 𝑣 𝑤

𝑒₂

𝑒3

𝑒₄ 𝑒5

𝑒6

𝑒₁

with𝐸⁰= {𝑢, 𝑣, 𝑤}and𝐸¹ = {𝑒₁, 𝑒₂, 𝑒₃, 𝑒₄, 𝑒₅, 𝑒₆}.

Consider the groupoid𝐺 with unit space 𝐺⁽⁰⁾ = {𝑢, 𝑣, 𝑤}and generators 𝑎, 𝑏, 𝑐where𝑑(𝑎) = 𝑢, 𝑡(𝑎) = 𝑑(𝑏) = 𝑣, 𝑑(𝑐) = 𝑡(𝑏) = 𝑤.

𝑢 𝑣 𝑤

𝑎

𝑏

𝑐

We define the action of𝐺by

𝑎 ⋅ 𝑒₁= 𝑒₂, 𝑎|_𝑒1 = 𝑢, 𝑎 ⋅ 𝑒₃= 𝑒₆, 𝑎|_𝑒3 = 𝑏, 𝑏 ⋅ 𝑒₂ = 𝑒₅, 𝑏|_𝑒2 = 𝑎, 𝑏 ⋅ 𝑒₆= 𝑒₄, 𝑏|_𝑒6 = 𝑐,