That comes from the general truth that the genitive absolute usually does not refer to anything in the independent clause; however, there are many exceptions, notably in the New Testament and in Koine.
ἔαροςéarosἀρχομένου,arkhoménou,οἱhoiἈθηναῖοιAthēnaîoiἔπλευσανépleusanεἰςeisΠροκόννησονProkónnēson[2] ἔαρος ἀρχομένου, οἱ Ἀθηναῖοι ἔπλευσαν εἰς Προκόννησονéaros arkhoménou, hoi Athēnaîoi épleusan eis Prokónnēson'When spring was beginning, the Athenians sailed to Proconnesus.
τὸνtònδὲdèΚῦρονKûronἦγονêgonἔσωésōοἱhoiθεράποντες,therápontes,κελεύσαντοςkeleúsantosτοῦtoûἈστυάγεοςAstuágeos[3] τὸν δὲ Κῦρον ἦγον ἔσω οἱ θεράποντες, κελεύσαντος τοῦ Ἀστυάγεοςtòn dè Kûron êgon ésō hoi therápontes, keleúsantos toû Astuágeos'The servants led Cyrus inside, after Astyages gave the order.
The perfect participle describes a situation which was already in existence and which still prevailed at the time of the action of the main verb, for example: ἤδηḗdēδ’d’ἐψηφισμένωνepsēphisménōnΘετταλῶν,Thettalôn,......ἧκενhêken[4] ἤδη δ’ ἐψηφισμένων Θετταλῶν, ... ἧκενḗdē d’ epsēphisménōn Thettalôn, ... hêken'Since the Thessalians had already voted ... he came back'The future is less often used in a genitive absolute.
An actual genitive absolute exists in German, such as klopfenden Herzens "(with) his/her heart beating", although its use is much less prominent compared to Greek (or to Latin's ablative or English's nominative in such constructions).