Gromov's pseudogroups and Tao's approximate groups In his article
Gromov, M. Almost flat manifolds. J. Differential Geom.  13  (1978), no. 2, 231–241
Gromov exploited a notion of a pseudogroup.  In his book
Tao, Terence. Hilbert's fifth problem and related topics. Graduate Studies in Mathematics, 153. American Mathematical Society, Providence, RI, 2014
Tao systematically developed the notion of an approximate group and gave several applications.
Both of these notions are in the subject that has come to be called geometric group theory.  Thus, Gromov builds his pseudogroup out of parts of the fundamental group with a suitable operation called Gromov product (by Buser and Karcher) which approximates composition of loops.  What is the precise relationship between the notion of pseudogroup and that of an approximate group?
 A: After some thinking, I no longer believe in any deep connection between the two notions. Their definitions sound similar and can indeed be used to establish some of the same results, yet the analogy seems superficial.
Let me record some references that might help those who want to think more.
A pseudogroup seems more of a technical device similar to thinking a space as the quotient of its universal cover by the deck-transformation action.
In the collapsing theory of manifolds with two sided curvature bounds a pseudogroup is a technical device allowing to desingularise the Gromov-Hausdorff convergence. To see how it is done it is most efficient to glance through the following:

*

*pp. 33-34 of Fukaya's book "Metric Riemannian geometry" https://www.math.kyoto-u.ac.jp/preprint/2004/16fukaya.pdf.


*p.494 of Lott's paper "Dimensional reduction and the long-time behavior of Ricci flow", https://math.berkeley.edu/~lott/2010-85-03-01.pdf.
Let me informally summarize what happens in the above references: a collapsing sequence of metric balls can be (under some curvature assumptions) realized as quotients of balls in the tangent spaces by action of pseudogroups. The actions subcoverge to an isometric action of a local Lie group on some Riemannian manifold. This is a convenient way to think about local collapse developed by Gromov and Fukaya.
(There are other ways to think about collapse that do not use the language of pseudogroups.)
Thus in this construction a pseudogroup approximates a local Lie group,
whose action gives a  description of the collapse.
On the other hand, an approximate group sits, by definition, in a local group, and sometimes captures  the algebraic properties of the associated global group, such as virtual nilpotence. To see how this applies to geometric problems look at the proof of Corollaries 11.13 in Breuillard-Green-Tao's "The structure of approximate groups", https://arxiv.org/abs/1110.5008. Here the geometric input is minimal (namely, the Bishop-Gromov volume comparison which gives a packing condition on the orbit of the fundamental group action in the universal cover).
The notion of an approximate group allows to decouple group theory and geometry, which is quite striking.
A: Here's working definition from What is... an Approximate Group? Let $A$ be a subset of a group with $A^{-1}=A$ then $A$ is an $K$-approximate group if $A^2$ is covered by a certain number $K$ of translates of $A$. 
These are not even subgroups or cosets.  


*

*Arithmetic sequences $ \{\sum n_i x_i : |n_i| < N_i \} $ is a $2^d$ approximate group.  It's not even closed under addition, it just kind of overlaps with itself.

*$S = \{\left( \begin{array}{ccc} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{array} \right) \text{ with }|a|, |c| \leq N \text{ and }|b| \leq N^2\}$ is an 100-subgroup if add it's inverses $A = S \cup S^{-1}$.  


It's not hard to come up with your own definition of approximate symmetry how about:
$$ A = \{ (a,b): \mathrm{gcd}(a,b) = 1 \text{ and } |a| ,|b| \leq N \} \subseteq \mathbb{Z}^2$$
and now rotate by an angle $a + bi \mapsto e^{i\theta}(a + bi)$.  This set has no intersection with itself.  Yet say the pairs of numbers $(a,b)$ which are relatively prime should be a rotationally symmetric set approximately  --- and this might not even fall under Tao/Green/Breullard's definition.  

The Structure of Approximate Groups outlines the types of almost-symmetry the have in mind:

A fair proportion of the subject of additive combinatorics is concerned
  with approximate analogues of exact algebraic properties, and the extent to which they resemble those
  algebraic properties. In this paper we are concerned with sets that are approximately closed under
  multiplication, which we do not necessarily assume to be commutative, and more specifically with
  approximate groups


In Almost Flat Structures Gromov is discussing differential geometry.  A pseudogroup is


*

*A set $\Gamma$ with a binary operation (that only works for some pairs) $a b$

*unique identity element $e$ and unique inverse $a^{-1}$

*$(ab)c = a(bc)$ (associativity)


In fact Gromov gives examples of his pseudogroup concept


*

*any symmetric subset $A \subseteq S_n$ containing the identity element $e \in A$ and $A = A^{-1}$

*his "local fundamental group" example places restrictions on his fundamental group so that the product is not always defined -- and he gave this structure a name.


These are closed under multiplication (when a multiplication exists) , while Approximate groups are definitely not closed.  Therefore, Gromov's definition is closer to that of groupoid and there is indeed something called the fundamental groupoid whose elements are paths and not loops.

To conclude there is no relation, but there are a few ways out:


*

*Cayley's Theorem shows every abstract group $G$ can be represented as permutations of a set.  Even infinite groups like $(\mathbb{R}, +)$.

*In Geometric Group Theory groups are thought of as fundamental groups of interesting spaces, such as trees or surfaces or 3-manifolds, or spaces which are not manifolds at all.
