This is the 16781st most frequent Russian word.
слуху
"Слуху" translates to "hearing" or "rumor" depending on context.
In this sentence, 'слуху' refers to 'rumor', in the dative case, illustrating reliance upon information spread by word of mouth.
Я доверился слуху, который был подтверждён многими людьми.
I trusted the rumor that was confirmed by many people.
Here, 'слуху' is in the dative case connected to 'тонким', referring to the sense of hearing, emphasizing auditory perception.
Человек с очень тонким слухом услышал звук на расстоянии миллиметров.
A person with very sharp hearing heard the sound at a distance of millimeters.
In this context, 'слуху' again refers to 'rumor', used abstractly to indicate focus on social information or stories.