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In this paper, we show that a surface group action on      S 1 {S^1}       with a non-zero Euler number has a positive topological entropy. We also show that if a surface group action on      S 1 {S^1}       has a Euler number which attains the maximal absolute value in the inequality of Milnor-Wood, then the topological entropy of the action equals the exponential growth rate of the group.

On topological entropy of group actions on 𝑆¹