  The combinatorial Loewner property was introduced by Bourdon and Kleiner as a
quasismmetrically invariant substitute for the Loewner property for general
fractals and boundaries of hyperbolic groups. While the Loewner property is
somewhat restrictive, the combinatorial Loewner property is very generic --
Bourdon and Kleiner showed that many familiar fractals and group boundaries
satisfy it. If $X$ is quasisymmetric to a Loewner space, it has the
combinatorial Loewner property. Kleiner conjectured in 2006 that the converse
to this holds for self-similar fractals -- the hope being that this would lead
to the existence of many exotic Loewner spaces. We disprove this conjecture and
give the first examples of spaces which are self-similar and combinatorially
Loewner and which are not quasisymmetric to Loewner spaces.
  In the process we introduce a self-similar replacement rule, called a linear
replacement rule, which is inspired by the work of Laakso. This produces a new
rich class of fractal spaces, where closed form computations of potentials and
their conformal dimensions are possible. These spaces exhibit a rich class of
behaviors from analysis on fractals in regards to diffusions, Sobolev spaces,
energy measures and conformal dimensions. These behaviors expand on the known
examples of Cantor sets, gaskets, Vicsek sets, and the often too difficult
carpet-like spaces. Especially the counter examples to Kleiner's conjecture
that arise from this construction are interesting, since they open up the
possibility to study the new realm of combinatorially Loewner spaces that are
not quasisymmetric to Loewner spaces.
