This is the 8638th most frequent Russian word.
отбывая
"Отбывая" means "serving" or "departing," depending on context.
In this sentence, "отбывая" is a gerund that indicates an ongoing action related to serving a prison sentence.
Он потерял контакт с родными, отбывая срок в тюрьме.
He lost contact with his relatives while serving time in prison.
Here, "отбывая" functions as a gerund describing the act of undergoing their punishment in a playful environment.
Дети веселились, отбывая наказание в игровой комнате.
The children were having fun while carrying out their punishment in the playroom.
In this case, "отбывая" acts as a gerund describing the act of fulfilling her military service during which she was writing letters.