There’s an option somewhere in Telegram’s settings to switch between GCM or their own push notifications service (I can’t find it in the FOSS version available at F-Droid).
Any app that needs to have a service running all the time, like push notifications, need to show a permanent notification for it. If you prefer the Telegram push service, you could tap and hold the notification and reduce the notification “importance level” to the minimum (don’t know the English naming). That way it’s only shown when you swipe down to show the notification drawer.