  We give an algorithmic and lower-bound framework that facilitates the
construction of subexponential algorithms and matching conditional complexity
bounds. It can be applied to intersection graphs of similarly-sized fat
objects, yielding algorithms with running time $2^{O(n^{1-1/d})}$ for any fixed
dimension $d \geq 2$ for many well known graph problems, including Independent
Set, $r$-Dominating Set for constant $r$, and Steiner Tree. For most problems,
we get improved running times compared to prior work; in some cases, we give
the first known subexponential algorithm in geometric intersection graphs.
Additionally, most of the obtained algorithms are representation-agnostic,
i.e., they work on the graph itself and do not require the geometric
representation. Our algorithmic framework is based on a weighted separator
theorem and various treewidth techniques.
  The lower bound framework is based on a constructive embedding of graphs into
d-dimensional grids, and it allows us to derive matching
$2^{\Omega(n^{1-1/d})}$ lower bounds under the Exponential Time Hypothesis even
in the much more restricted class of $d$-dimensional induced grid graphs.
