Learn Spanish (Latin America): Acabamos de pedir el permiso para que nos garanticen el espacio. - We just requested the permission so that they guarantee us the space.

The phrase para que (so that) introduces a purpose clause. In Spanish, para que always requires the subjunctive because it expresses a goal or an intended outcome, which is not yet a reality. Here, garanticen is the present subjunctive form of garantizar (to guarantee).

Could I just say para garantizarnos instead of para que nos garanticen?

You would only use para + infinitive (like para garantizarnos) if the subject doing both actions is the same. Here, we are requesting the permission, but they are guaranteeing the space. Because there is a change in subject between the two halves of the sentence, you must use para que followed by the subjunctive.

Why is garanticen spelled with a c instead of a z like in its infinitive form garantizar?

This is a mandatory spelling rule in Spanish. When a verb ends in -zar, the z changes to a c before the letter e. Since the present subjunctive endings for -ar verbs start with an e, the base of garantizar becomes garantic- before adding the -en ending for the "they" form.

I know usually means "to finish." Why does it mean "We just requested" here?