1. Introduction

The notion of soficity for groups was introduced by Gromov [MR1694588] in his work on symbolic dynamics. In 2010, Elek and Lippner [MR2566316] introduced soficity for equivalence relations in the same spirit as Gromov's original definition, i.e., an equivalence relation $R$, induced by some action of the free group $\mathbb{F}_\infty$, is sofic if the Schreier graph of the $\mathbb{F}_\infty$-space $X$ can be approximated, in a suitable sense, by Schreier graphs of finite $\mathbb{F}_\infty$-spaces.

Alternative definitions by Ozawa [ozawasoficnotes] and Pǎunescu [MR2826401] describe soficity at the level of the so-called full semigroup of $R$, which can be immediately generalized to groupoids. We will describe general elementary techniques to deal with (abstract) sofic groupoids.

1.1. Probability measure-preserving groupoids and full semigroups

We will follow the notations of [MR3130315]: Given a groupoid $G$, the source and range maps will be, respectively, $s(g)=g^{-1}g$ and $r(g)=gg^{-1}$ for $g\in G$, and the unit space of $G$ will be denoted $G^{(0)}$.

A discrete measurable groupoid is a groupoid $G$ endowed with a standard Borel space structure such that the product and inversion maps are Borel, and such that $s^{-1}(x)$ is countable for every $x\in G^{(0)}$.

The Borel full semigroup of a discrete measurable groupoid $G$ is the set $[[G]]_B$ of Borel subsets $\alpha\subseteq G$ such that the restrictions $s|_\alpha$ and $r|_\alpha$ of the source and range maps are injections, and thus Borel isomorphisms onto their respective images ([MR1321597]).

$[[G]]_B$ is an inverse monoid with the usual product and inverse of sets, namely

\begin{equation*} \alpha\beta=\left\{ab:(a,b)\in (\alpha\times\beta)\cap G^{(2)}\right\},\qquad \alpha^{-1}=\left\{a^{-1}:a\in\alpha\right\} \end{equation*}
and $G^{(0)}$ is the unit of $[[G]]_B$, which we will instead denote by $G^{(0)}=1$ when no confusion arises.

A probability measure-preserving (pmp) groupoid is a discrete measurable groupoid $G$ with a Borel probability measure $\mu$ on $G^{(0)}$ satisfying $\mu(s(\alpha))=\mu(r(\alpha))$ for all $\alpha\in[[G]]_B$. We write $(G,\mu)$ for a pmp groupoid when we need the measure $\mu$ to be explicit. The measure $\mu$ induces a pseudometric $d_{\mu}$ on $[[G]]_B$ via

\begin{equation*} d_{\mu}(\alpha,\beta)=\mu(s(\alpha\triangle\beta))=\mu(r(\alpha\triangle\beta)). \end{equation*}

The trace of $\alpha\in[[G]]_B$ is defined as $\operatorname{tr}(\alpha)=\mu(\alpha\cap G^{(0)})$. In fact, the trace and the pseudometric above, along with the semigroup operation, determine each other: For example, the unit $1$ of $[[G]]_B$ is the only element of trace $1$, and

\begin{equation*} d_\mu(\alpha,\beta)=\operatorname{tr}(\alpha^{-1}\alpha)+\operatorname{tr}(\beta^{-1}\beta)-\operatorname{tr}(\alpha^{-1}\alpha\beta^{-1}\beta)-\operatorname{tr}(\beta^{-1}\alpha) \end{equation*}
and similarly one can write the trace in terms of the pseudometric $d_\mu$.

The (measured) full semigroup of a pmp groupoid $(G,\mu)$ is the quotient metric space $[[G]]$ (or $[[G]]_\mu$ to make $\mu$ explicit) of $[[G]]_B$ under the pseudometric $d_\mu$. The semigroup operation and the trace factor through $[[G]]$, which determines its (canonical) semigroup structure.

The Borel full group $[G]_B$ of a discrete measurable groupoid $G$ is the set of those $\alpha\in[[G]]_B$ with $s(\alpha)=r(\alpha)=G^{(0)}$, and, when $G$ is pmp, the image of $[G]_B$ in $[[G]]$, denoted $[G]$ or $[G]_\mu$, is called the (measured) full group of $G$.

Definition 1.1.

A subset $A$ of a pmp groupoid $(G,\mu)$ is called null if $\mu(s(A))=0$ (equivalently, $\mu(r(A))=0$), and conull if its complement $G\setminus A$ is null. A property of the points of $G$ is said to hold a.e. (almost everywhere) if it holds on a conull subset.

Example 1.2.

Let $R$ be a countable Borel equivalence relation on a standard probability space $(X,\mu)$, and suppose $\mu$ is invariant (see [MR0578656]). We can see $R$ as a pmp groupoid as follows: the product is defined by $(x,y)(y,z)=(x,z)$. The unit space of $R$ is the diagonal $\left\{(x,x):x\in X\right\}$, which we identify with $X$ and endow with the probability measure $\mu$. The Borel full semigroup of $R$ can be identified with the semigroup of partial Borel isomorphisms $f:A\to B$, $A,B\subseteq X$, for which $(f(x),x)\in R$ for all $x\in A$, by associating such $f$ to the inverse of its graph, $\left\{(f(x),x):x\in X\right\}$. The pmp groupoids which are isomorphic (in the measure-theoretic sense) to one constructed this way are called principal groupoids.

Example 1.3.

Let $Y$ be a finite set and $Y^2$ the largest equivalence relation on $Y$, endowed with the usual (discrete) Borel structure. The only probability measure on $Y$ which makes $Y^2$ pmp is the normalized counting measure: $\mu_\#(A)=|A|/|Y|$. We denote the associated metric by $d_\#$ and call it the normalized Hamming distance.

Note that if $Y$ and $Z$ are finite sets, then the map $[[Y^2]]\ni\alpha\mapsto \alpha\times(Z^2)^{(0)}\in[[Y^2\times Z^2]]$ is a trace-preserving embedding. The map $(y_1,y_2,z_1,z_2)\mapsto (y_1,z_1,y_2,z_2)$ is a measure-preserving isomorphism between the groupoids $Y^2\times Z^2$ and $(Y\times Z)^2$, which induces a trace-preserving isomorphism between the respective two full semigroups. Therefore if $Y$ and $Z$ are finite sets, there are a finite set $W$ and trace-preserving embeddings from $[[Y^2]]$ and $[[Z^2]]$ into $[[W^2]]$

Definition 1.4.

A sofic approximation of a pmp groupoid $G$ is a sequence of maps $\pi=\left\{\pi_k:[[G]]\to[[Y_k^2]]\right\}$, where $Y_k$ are finite sets, such that for all $\alpha,\beta\in[[G]]$,

  1. $\lim_{k\to\infty}\operatorname{tr}(\pi_k(\alpha))=\operatorname{tr}(\alpha)$;

  2. $\lim_{k\to\infty}d_\#(\pi_k(\alpha\beta),\pi_k(\alpha)\pi_k(\beta))=0$.

A pmp groupoid $G$ is sofic if it admits a sofic approximation.

We'd like to note that this definition of sofic approximation differs from that of Bowen [MR3286052], where aditional properties are required of the sequence $\pi$ in order to study the entropy of sofic groupoids. However, the existence of a sofic approximation in the stronger sense of [MR3286052] is actually equivalent to the existence of a sofic approximation as in the definition above.