By gears, maybe?
Incidentally, what makes you think the shaft of a turbine engine is any hotter than the crankshaft of a piston engine? Piston engines get pretty hot inside, too - but the shaft protudes outside the engine, and there is a pulley bolted typically onto the front of the engine. Can't do this with a turbine engine, since the air-flow around the front and rear of the engine is essential - I'd have thought that's why there's no belt-drive. That and the fact that the turbines turn much faster than most piston engines.
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