dramatic question

I’m puzzled by a line in the Bacchae, 1223:

(Kadmos)
ἤδη κατ’ ἄστυ τειχέων ἔσω βεβὼς

The sense of it’s clear enough to me. But I can’t tell why the two nouns are as they are, especially why τείχη is genitive. Any ideas?

?σω can take the genitive when it means “inside”(see Middell-Liddell), so τειχέων is dependent on that. κατά with the accusative can mean “throughout.” Without any context, which could change the translation, I would say it means something like, "Having marched inside the walls throughout the city. If you haven’t read it already, u should check out Martha Nussbaum’s introduction to C.K. Williams translation of the Bacchae. Amazing interpretation of the Bacchae in light of Nietzsche and other interpreters.

Thanks for the help and the recommendation! In retrospect that seems rather obvious, and I’d even looked up all the words in my puzzlement, but somehow overlooked ἔσω taking the genitive (which I’ve surely seen before).