You can take advantage of the files maintained in ~/.gitolite/logs
by gitolite.
Those log files will record any git operation, authorized or denied.
But they aren't "exported" or visible outside the gitolite server by default.
To expose them, you need some kind of audit log web server, like for instance logstash
(which is a generic tool, for managing events and logs: it isn't linked to gitolite, but can parse its logs and display information about their content).