It is a popular belief that Hanuman comes to the place where Ramayana is getting preach and stays till the end.
Is that belief true?
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That belief is true. There is a mangala shloka (invocatary verse) written by Goswami Tulsidas. Translation from greensmesg.org:
yatra yatra raghunāthakīrtanaṃ tatra tatra kṛtamastakāṃjalim |
vāṣpavāriparipūrṇālocanaṃ mārutiṃ namata rākṣasāntakam ||
Wherever the glories of Raghunatha are sung, there, with hands held over His bowed head in salutation, ... and eyes filled with tears, Maruti (Bhakta Hanuman) is present; I salute Maruti who puts an end to the Rakshasas.
Even in the religious discources where Ramcharitmanas is said, a seat is left empty so that Lord Hanuman visits that place and hears to Ramayana. It is not because he doesn't know Ramayana but he loves the glories of Rama.
It is how Goswami Tulsidas met Hanuman. Wikipedia page on Tulsidas gives related info. From Darshan of Hanuman:
That evening Tulsidas noted that the first listener to arrive at his discourse was an old leper, who sat at the end of the gathering. After the Katha was over, Tulsidas quietly followed the leper to the woods. In the woods, at the spot where the Sankat Mochan Temple stands today, Tulsidas firmly fell at the leper's feet, shouting "I know who you are" and "You cannot escape me". At first the leper feigned ignorance but Tulsidas did not relent. Then the leper revealed his original form of Hanuman and blessed Tulsidas. When granted a boon, Tulsidas told Hanuman he wanted to see Rama face to face. Hanuman told him to go to Chitrakuta where he would see Rama with his own eyes.
This story is a famous one. Moreover, the above Wikipedia page cites multiple references for each claim made in the story of Tulsidas. So, we can say that the belief Lord Hanuman stays at a place where Ramayana is sung or Raghunatha's glories are sung as true.
The belief seems to have some connection with these verses of Uttara Kanda of Valmiki Ramayana:
Kakuthstha [Rama] said to Hanuman "It is settled that thou shalt live forever, do thou, now observe thy promise. As long as my history shall run current in this world do thou at my command live happily." Being thus addressed by the high souled Raghava, Hanuman attained to great delight and said: As long as the sacred theme shall pass current in this world I shall live here carrying out thy commands.