In the spirit of peripheral subgroups in relatively hyperbolic groups, we exhibit a simple
class of quasi-isometrically rigid subgroups in graph products of finite groups, which we
call
eccentric subgroups. As an application, we prove that if two right-angled Coxeter
groups
and
are quasi-isometric, then for any minsquare subgraph
, there exists a minsquare
subgraph
such that the
right-angled Coxeter groups
and
are quasi-isometric as well. Various examples of non-quasi-isometric groups
are deduced. Our arguments are based on a study of nonhyperbolic Morse
subgroups in graph products of finite groups. As a by-product, we are able to
determine precisely when a right-angled Coxeter group has all its infinite-index
Morse subgroups hyperbolic, answering a question of Russell, Spriano and
Tran.
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