It's usually just a nylon fabric, not much different from what a windbreaker jacket would be made of. Infinity Lames use an absorptive, open fabric that wicks sweat away-- it's not technically to the letter of the rules (since it has holes in it), but SEMI has given it their blessing.
Some really, really old-- like 1960's or earlier vintage-- lames, may have a fairly thick vinyl lining. If you've ever fenced with a relay-based box from that era (which sourced a rather higher current through the weapons than modern boxes), you'll understand the rationale.
-Dave
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