Given any space X, the free degree of homeomorphisms on \(X\) is the minimum positive integer \(n\) such that for any homeomorphism \(h\), at least one of the iterates \(h\), \(h^2\), \(\ldots\), \(h^n\) has a fixed point. This is a classical concept which can be traced back to the work of J. Nielsen in the 1940’s [Mat. Tidsskr. B 1942, 25–41 (1942; Zbl 0027.09602)]. A relatively recent work was given by S. Wang [Topology Appl. 46, No. 1, 81–87 (1992; Zbl 0757.55004)].
The authors of this paper obtain an upper bound for the free degree of connected compact surfaces, which depends only on the genus. The upper bound is \(24g-24\) for orientable surfaces of genus \(g\geq 2\), and is \(12g -24\) for non-orientable surfaces of genus \(g\geq 3\). The methods in this paper are based on Nielsen fixed point theory and Thurston’s theorem about classification of homeomorphisms on surfaces. The former is used to detect the fixed point of iterates of given homeomorphism, while the latter allows an induction based on genus and number of boundary components. The key point is the argument about the non-vanishing of the Nielsen numbers of standard homeomorphisms on surfaces introduced by B. Jiang and J. Guo in [Pac. J. Math. 160, No.1, 67–89 (1993; Zbl 0829.55001)].