Learn Russian: Они опасались независимости нового филиала. - They feared the independence of the new branch.

Verbs expressing fear, apprehension, or avoidance, like опасаться (to fear or be wary of), require their object to be in the Genitive case, not the Accusative. This belongs to a broader category of verbs dealing with deprivation, avoidance, or reaching for a goal that take the Genitive.

Is there a difference in meaning between опасались and боялись?

Yes. While both can be translated as 'feared', бояться is the standard word for being afraid, often describing a direct, emotional fear. Опасаться is slightly more formal and means to be apprehensive or wary of a potential risk. In a business context like this, опасаться is a very natural choice.

Независимости ends in -и. Is it plural here?

No, it is singular. Независимость (independence) is a feminine noun that ends in a soft sign (-ь), meaning it belongs to the 3rd declension. For these nouns, the Genitive singular ending is -и.

Why is нового филиала also in the Genitive case?

This creates a chain of Genitives, which is very common in Russian. While независимости is in the Genitive because it is the object of the verb опасались, нового филиала is in the Genitive to show possession or relation, translating to 'of the new branch'.