Is there any etymological or grammatical relationships between γένετ᾽ and ἄρχω in ancient Greek?
I find that perseus, for the noun ἀρχή, does show γενέσθαι. I know that γένετ᾽ is doric aeolic and ἀρχή is attic ionic but I wonder if there are any indications that ἀρχή may find a common stem in doric aeolic dialects? The meanings are somewhat different in "come into a new state of being" and "beginning, origin" but it would seem there is some common ground somewhere perhaps in the meanings.
ἀρχή , ἡ, (v. ἄρχω)
A. beginning, origin, “νείκεος ἀ.” Il.22.116; “πήματος” Od.8.81; “φόνου” 21.4, etc.; opp. τέλος, Hdt.7.51, etc.; opp. τελευτή, Thgn.607, cf. Pl.Lg.715e, Hp.Morb.1.1; “ἀ. γενέσθαι κακῶν”
thank you for your question. The words you mentioned are not connected etymologically at all. Their meanings, though, "to become" and "to start" - as you also mentioned - are not so far away, but it is very clear what each of them means; and they cannot replace each other, I would say.
In Perseus' definition I don't see anything that supports a possible connection. It includes examples of usages from random sources. I will re-submit it by adding coloured brackets in order to clarify how it has to be considered (dictionaries use short forms to save up space, as you probably know). Please, find the definition as attached, because I am experiencing display problems with the polytonic script at the moment. Apologies for that.
Please, let me know if this makes things clear.