For every group
,
we define the set of
hyperbolic structures on
, denoted
by ,
which consists of equivalence classes of (possibly infinite) generating sets of
such
that the corresponding Cayley graph is hyperbolic; two generating sets of
are
equivalent if the corresponding word metrics on
are bi-Lipschitz
equivalent. Alternatively, one can define hyperbolic structures in terms of cobounded
–actions
on hyperbolic spaces. We are especially interested in the subset
of
acylindricallyhyperbolic structures on
,
ie hyperbolic structures corresponding to acylindrical actions. Elements of
can be
ordered in a natural way according to the amount of information they provide about the
group
.
The main goal of this paper is to initiate the study of the posets
and
for various
groups .
We discuss basic properties of these posets such as cardinality and
existence of extremal elements, obtain several results about hyperbolic
structures induced from hyperbolically embedded subgroups of
, and
study to what extent a hyperbolic structure is determined by the set of loxodromic
elements and their translation lengths.
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