1) to be leader, to lead the way
2) to rule, command
2a) of a province, to be governor of a province
2b) said of a proconsul, of a procurator
Part of Speech: verb
From G2232; to act as ruler: - be governor.
These two sites give similar information, with the definition from several dictionaries and statistics on the use of the word.
Perseus Digital Library
University of Chicago's Logion lexicon
Thayer's dictionary plus other information.
From this site's dictionary (in Italian)
Click on the first column to search for that word as a form of the root ἡγεμονεύω; click on the second column to search for that grammatical form of the root ἡγεμονεύω; click on the third column to search for that word and grammatical form; click on the total to search for the root ἡγεμονεύω.