Moisés Silva, in discussing ἒργα νόμου (law-works, or works of law) and πίστις Χριστοῦ (Christ-faith, or faith in/of Christ) has some helpful words about the limitations of grammar in interpreting genitival constructions:
The grammatical case tells us virtually nothing except that there is some kind of unspecified relationship between the two nouns in the construction.
He shows how this works in English:
The corresponding phenomenon in English is the mere juxtaposition of two nouns, such as
spring picnic (an activity that takes place in the spring),
car sale (an event in which someone sells cars [at a special price],
brick house (a building constructed with bricks),
house furniture (furnishings intended for [or found in] a household),
church history (the course of events that have taken place within the context of [or have been produced by or have otherwise affected] the church), etc.
In these cases the first noun (which corresponds to the genitival noun in Greek) functions adjectivally and modifies the second noun.
There is nothing about the grammar, of even about the uses of such a syntactical construction, that tells us how to understand the relationship between the two nouns.
How then do we interpret genitival constructions if grammar is insufficient?
Our interpretation depends rather on two other facts:
(a) our lexical knowledge (esp. of the first noun), and
(b) our contextual or historical knowledge.
Thus we can say that spring picnic entails a “temporal” relationship because
(a) we know that spring designates a period of time and
(b) we know that in our culture picnics are often held during the spring season.
It is unlikely, but not impossible, that spring picnic might refer to, say, an event held in the winter but having spring as its theme (cf. Christmas concert, which does not necessarily take place on Christmas day but celebrates the significance of that day). Should this second sense be intended, we would be able to interpret the phrase correctly only if someone told us specifically about such a peculiar picnic or if we have attended similar events in the past.
The entire essay is worth reading, and to my mind persuasive. See Moisés Silva, “Faith Versus Works of Law in Galatians,” in Justification and Variegated Nomism: The Paradoxes of Paul (ed. D. A. Carson, Peter T. O’Brien, and Mark A. Seifrid; WUNT 181; Grand Rapids: Baker, 2004), 217-48. The quotes above are from p. 220.