Each Cantor set C in S3 has for complement an open 3-manifold M3 with end set C. Properties of the embedding of the Cantor set give rise to properties of the corresponding complementary 3-manifold M3. See [Souto and Stover 2013], [Garity and Repovš 2013], and [Garity et al. 2014] for examples of this. We investigate possible group actions on the end set C of the open 3-manifold M3 in the following sense: the homogeneity group of the end set is the group of homeomorphisms of the end set C that extend to homeomorphisms of the open 3-manifold M3. Referring specifically to the embedding of the Cantor set, this group can also be called the embedding homogeneity group of the Cantor set. See [Dijkstra 2010] and [van Mill 2011] for some other types of homogeneity.
[Antoine 1920] L. Antoine, “Sur la possibilité d’étendre l’homéomorphie de deux figures à leur voisinage”, C. R. Acad. Sci. (Paris)171 (1920), 661–663. Zbl 47.0524.01 [Daverman 1979] R. J. Daverman, “Embedding phenomena based upon decomposition theory: wild Cantor sets satisfying strong homogeneity properties”, Proc. Amer. Math. Soc. 75:1 (1979), 177–182. MR 80k:57031 Zbl 0407.57011 [Dickman 1968] R. F. Dickman, Jr., “Some characterizations of the Freudenthal compactification of a semicompact space”, Proc. Amer. Math. Soc. 19 (1968), 631–633. MR 37 #884 Zbl 0157.29404 [Dijkstra 2010] J. J. Dijkstra, “Homogeneity properties with isometries and Lipschitz functions”, Rocky Mountain J. Math. 40:5 (2010), 1505–1525. MR 2011j:54033 Zbl 1227.54030 [Freudenthal 1942] H. Freudenthal, “Neuaufbau der Endentheorie”, Ann. of Math. (2) 43 (1942), 261–279. MR 3,315a Zbl 0060.40006 [Garity and Repovš 2013] D. J. Garity and D. Repovš, “Inequivalent Cantor sets in R3 whose complements have the same fundamental group”, Proc. Amer. Math. Soc. 141:8 (2013), 2901–2911. MR 3056580 Zbl 06184489 [Garity et al. 2005] D. J. Garity, D. Repovš, and M. Željko, “Uncountably many inequivalent Lipschitz homogeneous Cantor sets in R3”, Pacific J. Math. 222:2 (2005), 287–299. MR 2006m:54056 Zbl 1116.54015