We introduce the concept of Hofer-Zehnder G-semicapacity (or G-sensitive Hofer-Zehnder capacity) and prove that given a geometrically bounded symplectic manifold (M,ω) and an open subset N⊂M endowed with a Hamiltonian free circle action φ
Nearby in the stack
then
N
has bounded Hofer-Zehnder
Gφ
-semicapacity, where
Gφ⊂π1(N)
is the subgroup generated by the homotopy class of the orbits of
φ
. In particular,
N
has bounded Hofer-Zehnder capacity. We give two types of applications of the main result. Firstly, we prove that the cotangent bundle of a compact manifold endowed with a free circle action has bounded Hofer-Zehnder capacity. In particular, the cotangent bundle
T∗G
of any compact Lie group
G
has bounded Hofer-Zehnder capacity. Secondly, we consider Hamiltonian circle actions given by symplectic submanifolds. For instance, we prove the following generalization of a recent result of Ginzburg-Gürel: almost all low levels of a function on a geometrically bounded symplectic manifold carry contractible periodic orbits of the Hamiltonian flow, provided that the function attains its minimum along a closed symplectic submanifold.