Group associated to Isometries of Space. Reverse direction.

Question

I have seen several posts seeking to find groups of isometries associated to several metric spaces or figures/subspaces embedded in them. I am trying to go in the opposite direction and, given a group $G$, to find the figures for which it describes the isometries. I think this is somehow related to Cayley graphs and Frucht's theorem , but I am having trouble describing and defining the assignment more precisely. Any examples? If we were given, e.g., an alternating group or some other group, how would we find the spaces for which $G$ is the group of isometries?

Answer 1

Yes, indeed, this is a generalization of Frucht's theorem, discussed in this Wikipedia artcile:

1. Frucht's Theorem. For every finite group $G$ there exists a finite, simplicial, connected graph $\Gamma=\Gamma_G$ such that the group of graph-automorphisms of $\Gamma$ is isomorphic to $G$,

Robert Frucht, Graphs of degree three with a given abstract group, Can. J. Math. 1, 365-378 (1949). ZBL0034.25802.

(Here a graph is simplicial or simple if it contains no monogons and digons, i.e. is a 1-dimensional simplicial complex.)

2. Frucht's theorem was generalized, independently, by de Groot (his paper is mostly about homeomorphisms, the result about grap-automorphisms is in Theorem 6) and Sabidussi:

J. de Groot, Groups represented by homeomorphism groups. I, Math. Ann. 138, 80-102 (1959). ZBL0087.37802.

Gert Sabidussi, Graphs with given group and given graph-theoretical properties, Can. J. Math. 9, 515-525 (1957). ZBL0079.39202.

For every connected graph $\Gamma$ one defines its graph-metric $d_\Gamma$: The distance between vertices is the combinatorial length of the shortest edge-path between these vertices. By declaring every edge to be isometric to the unit interval, one extends this metric to the geometric realization $|\Gamma|$ of $\Gamma$ (the underlying 1-dimensional topological space) so that $Aut(\Gamma)\cong \mathrm{Isom}(|\Gamma|, d_\Gamma)$ (provided that $\Gamma$ is simplicial), except in the (easily avoidable) cases when $|\Gamma|$ is either homeomorphic to ${\mathbb R}$ or $S^1$. Hence:

Theorem. For every abstract group $G$ there exists a metric space $X$ whose full isometry group is isomorphic to $G$.

3. One can ask for various refinements of this theorem, e.g. realizing the given (Hausdorff) topological group $G$ as the group of isometries of the given metric space (equipped with the compact-open topology) or realizing a given Lie group as the full isometry group of a Riemannian manifold. It is known, for instance, that:

Theorem. Every Polish group is isomorphic to the isometry group of some separable complete metric space,

Su Gao and Alexander Kechris, On the classification of Polish metric spaces up to isometry, Mem. Am. Math. Soc. 766, 78 p. (2003). ZBL1012.54038.

There are partial results regarding isometry groups of Riemannian manifolds, e.g.

Theorem. Every discrete countable group is isomorphic to the fundamental group of a Riemannian manifold,

Jörg Winkelmann, Realizing countable groups as automorphism groups of Riemann surfaces, Doc. Math. 6, 413-417 (2001). ZBL0989.30030.

However, for general Lie groups, I do not think there is a definitive answer.