  Sofic groups generalise both residually finite and amenable groups, and the
concept is central to many important results and conjectures in measured group
theory. We introduce a topological notion of a sofic boundary attached to a
given sofic approximation of a finitely generated group and use it to prove
that coarse properties of the approximation (property A, asymptotic coarse
embeddability into Hilbert space, geometric property (T)) imply corresponding
analytic properties of the group (amenability, a-T-menability and property
(T)), thus generalising ideas and results present in the literature for
residually finite groups and their box spaces. Moreover, we generalise coarse
rigidity results for box spaces due to Kajal Das, proving that coarsely
equivalent sofic approximations of two groups give rise to a uniform measure
equivalence between those groups. Along the way, we bring to light a coarse
geometric view point on ultralimits of a sequence of finite graphs first
exposed by Jan \v{S}pakula and Rufus Willett, as well as proving some bridging
results concerning measure structures on topological groupoid Morita
equivalences that will be of interest to groupoid specialists.
