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compute the least acceleration with which a man of mass 75 kg can slide down a rope which can sustain a weight of 580 N

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You really mean the minimum deceleration. The man's weight is M g = 735 N. For the man to decelerate at a rate a, the rope must apply a friction force F to the man such that

M g - F = M a

F is also the rope tension, which must not exceed 580 N.

At maximum possible F (580 N), one gets the minimum possible acceleration, a_min

a_min = g - 580/M = 9.8 - (580/75)
= 2.1 m/s^2
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