So I'm minding my business and I get this phone call from a friend and fellow literature student. A good friend of mine, she tells me she's having some trouble with Ancient Greek. Her problem is that some professor mentioned the genitive absolute just in passing. She was afraid it would come up in the next test and she had no idea how it worked. I tried to explain her as best as I could what it was about, comparing with the ablative absolute in Latin. To me it appears clear from examples:
τῶν ἀνδρῶν πολεμούντων, αἱ γυναῖκες μόναι οἴκοι εἰσίν
While the men are waging war, the women are at home by themselves.
It implies two actions that are done simultaneously. It seems greeks didn't like to use coordinating words, so they came up with this kind of constructions. This is even present in such examples as the aorist participle in the words of Leonidas: Μολὼν λαβέ, "having come, take", for a simpler "Come and take them".
The thing is... how do you recognize said genitive? Well at first glance when a sentence begins with a genitive, but this is not so. The best way I could think of recognizing it is when you see an independent genitive with a participle. That to me is the best way to look at it.
But hey, beauty is in the eye of the beholder, plus eventually this didn't even come in the exam.