  Classical persistent homology is a powerful mathematical tool for shape
comparison. Unfortunately, it is not tailored to study the action of
transformation groups that are different from the group Homeo(X) of all
self-homeomorphisms of a topological space X. This fact restricts its use in
applications. In order to obtain better lower bounds for the natural
pseudo-distance d_G associated with a subgroup G of Homeo(X), we need to adapt
persistent homology and consider G-invariant persistent homology. Roughly
speaking, the main idea consists in defining persistent homology by means of a
set of chains that is invariant under the action of G. In this paper we
formalize this idea, and prove the stability of the persistent Betti number
functions in G-invariant persistent homology with respect to the natural
pseudo-distance d_G. We also show how G-invariant persistent homology could be
used in applications concerning shape comparison, when the invariance group is
a proper subgroup of the group of all self-homeomorphisms of a topological
space. In this paper we will assume that the space X is triangulable, in order
to guarantee that the persistent Betti number functions are finite without
using any tameness assumption.
