Perseus analyses of γαίων:

γαΐων,γάιος (on land): masc/fem/neut gen pl (doric)
γαίω (rejoice, exult): pres part act masc nom sg
γαιόω (make land, make solid): imperf ind act 3rd pl (doric aeolic)
γαιόω (make land, make solid): imperf ind act 1st sg (doric aeolic)

LSJ entries

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γάϊος

γάϊος[ᾱ], α , ον, Dor.for γήϊος,

A. on land, A. Supp. 826 (lyr.); earthy, γ. κόνις Id. Th. 736; of the land, κόγχοι Epich. 42.9; παῖς γ. child of earth, terrae filius, of a slave, prob. in IG 14.1432 (cf. γάϊος παρὰ Ἰταλιώταις καὶ Ταραντίνοις ὁ μίσθιος Eust. 188.30, cf. EM 223.24); ἄνεμος a land wind, Hsch.; also, = ἐργάτης βοῦς, Id., EM l.c.

II. τὸν γάϊον, = καταχθόνιον, prob. in A. Supp. 156 (lyr.).

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γαίω

γαίω, impf. γαίεσκον Hsch.:—

Α. rejoice, exult, Hom. only in Il., in phrase, κύδεϊ γαίων Il. 1.405, 5.906, 8.51; [Σφαῖρος] μονίῃ γαίων Emp. 27.4. (γάϝ -y ω , v. γάνυμαι.)

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γαιόω

A. make land, make solid, Tz. H. 1.907, al.