In Nielsen fixed point theory, the fixed point set of a self-map is considered as a disjoint union of fixed point classes: two fixed points are in the same class if and only if they can be joined by a path (called Nielsen path) which is a path homotopic (relative to endpoints) to its own image. Each fixed point class \(F\) of a self-map \(f\) has an integer-valued index \(ind(f, F)\). The sum of the indices of all fixed point classes is just the Lefschetz number \(L(f)\) and the Nielsen number \(N(f)\) is the number of fixed point classes with non-zero indices.
B. Jiang [Math. Ann. 311, No. 3, 467–479 (1998; Zbl 0903.55003)] showed that for a surface \(S\) with negative Euler characteristic, there is a constant \(B\) such that \(|ind(f, F)|\leq B\) for any fixed point class \(F\) of any self-map \(f\). Now, people use the notation “bounded index property (BIP)”. Moreover, one can consider the “bounded index property for homeomorphisms (BIPH)”, indicating that there is a bound for the indices of all homeomorphisms.
The author of the paper under review proves that any compact hyperbolic manifold \(M\) has the BIPH. Moreover, if the dimension of \(M\) is at least \(4\), there is a precise inequality for the index bound: \(\max\{N(f), |L(f)|\}\leq \sum_{F}|ind(f,F)|\leq \max\{\dim H_*(M; \mathbb Z_p)\mid p \text{ is prime} \}\) for any homeomorphism \(f:M\to M\). For \(\dim(M)=3\), the author also gives a concrete bound in some restricted cases. Of course, compact hyperbolic manifolds of dimension \(2\) are the cases considered by B. Jiang.