Apologies for the inescapably soft question.

Does there exist a theory that aims to develop tools analogous to those of group theory, except for the study of objects that are merely almost symmetrical? I am not sure what exactly it would look like, and I haven't been able to find what I'm looking for via internet searches. Here are the types of objects of study I have in mind:

  • A convex polyhedron that is a slight perturbation of a regular icosahedron (perhaps I realized the icosahedron as the intersection of half-spaces, and moved one of the half-spaces a tiny bit) is not $A_5$-symmetrical, but e.g. there will be an action of $A_5$ on the ambient space such that all the images of my almost-regular icosahedron will be contained in a thin tubular neighborhood of its original position. If I know my object has this type of "almost symmetry" then I know something about it constraining its geometry, even though it is not actually symmetrical.

  • Consider a rooted tree consisting of a root with 10 branches, each of which has 100 leaves, except one which has 101. If all the branches had 100 leaves, then the tree would have an automorphism group isomorphic to $S_{100}\wr S_{10}$, acting transitively on the leaves. As it is, the automorphism group is $(S_{100}\wr S_9) \times S_{101}$, with 900 leaves falling into one orbit and 101 falling into the other. There's no room in these considerations for the fact that 100 and 101 are close-together numbers. In some contexts, this is clearly as it should be, but in others, one might want a more permissive way to capture the almost-symmetrical nature of the graph. For example, if the tree is the setting for some sort of game, and the players begin the game on the leaves, and move about the tree as the game progresses, then an automorphism group transitive on the leaves expresses a kind of "fairness" with respect to the initial positions of the players. In this context, depending on the rules of the game, it might be that the game in which one of the branches has 101 instead of 100 leaves is "still fair enough", and one would then want a more permissive description that allows to capture this "near equality of initial states" than is captured by the actual graph automorphism group.

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    $\begingroup$ How about bounded cohomology and group quasi-homomorphisms? $\endgroup$ – anomaly Jan 28 at 17:09
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    $\begingroup$ Are you familiar with the sorites paradox and related topics? It is thought provoking. $\endgroup$ – Somos Jan 28 at 17:34
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    $\begingroup$ How about quasi-isometries? They pop up all the time in geometric group theory, particularly in hyperbolic groups. $\endgroup$ – anomaly Jan 30 at 0:35
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    $\begingroup$ @anomaly - thank you, good call, this may be helpful! $\endgroup$ – Ben Blum-Smith Jan 30 at 0:39
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    $\begingroup$ Also check out approximate groups: en.wikipedia.org/wiki/Approximate_group. $\endgroup$ – Moishe Kohan Jan 30 at 0:41

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