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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?

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?