Cheeger isoperimetric constants of Gromov-hyperbolic spaces with quasi-poles (Q2707941)

The paper concerns the following problem: Given a complete Gromov-hyperbolic space \(X\) with bounded local geometry, find additional conditions under which a Cheeger constant \(h(X)\) becomes strictly positive. (Recall that \(h(X)\) is defined for Riemannian manifolds as the greatest lower bound for quotients \(|\partial D|/|D|\), where \(D\) ranges over all compact domains and \(|\cdot |\) denotes suitable Lebesgue measures; if \(X\) is a graph, then \(h(X)\) is the greatest lower bound for the quotients \(|\partial A|/|A|\), where \(A\) ranges over all non-empty finite sets of vertices of \(X\), \(\partial A\) consists of all the vertices distant \(1\) from \(A\) and \(|\cdot |\) denotes cardinality of sets.) The author shows that such conditions can be formulated as follows: (1) \(X\) has a quasi-pole and (2) the diameters of all the connected components of the ideal boundary \(X(\infty)\) admit a positive lower bound. (A quasi-pole is a compact subset \(K\) of \(X\) such that any point of \(X\) lies (for some \(c > 0\)) within distance \(\leq c\) from a geodesic ray originated in \(K\).) Suitable examples show that none of conditions (1)--(2) can be removed. Some further applications are given.