  In recent years, high dimensional expanders have been found to have a variety
of applications in theoretical computer science, such as efficient CSPs
approximations, improved sampling and list-decoding algorithms, and more.
Within that, an important high dimensional expansion notion is \emph{cosystolic
expansion}, which has found applications in the construction of efficiently
decodable quantum codes and in proving lower bounds for CSPs.
  Cosystolic expansion is considered with systems of equations over a group
where the variables and equations correspond to faces of the complex. Previous
works that studied cosystolic expansion were tailored to the specific group
$\mathbb{F}_2$. In particular, Kaufman, Kazhdan and Lubotzky (FOCS 2014), and
Evra and Kaufman (STOC 2016) in their breakthrough works, who solved a famous
open question of Gromov, have studied a notion which we term ``parity''
expansion for small sets. They showed that small sets of $k$-faces have
proportionally many $(k+1)$-faces that contain \emph{an odd number} of
$k$-faces from the set. Parity expansion for small sets could be used to imply
cosystolic expansion only over $\mathbb{F}_2$.
  In this work we introduce a stronger \emph{unique-neighbor-like} expansion
for small sets. We show that small sets of $k$-faces have proportionally many
$(k+1)$-faces that contain \emph{exactly one} $k$-face from the set. This
notion is fundamentally stronger than parity expansion and cannot be implied by
previous works.
  We then show, utilizing the new unique-neighbor-like expansion notion
introduced in this work, that cosystolic expansion can be made
\emph{group-independent}, i.e., unique-neighbor-like expansion for small sets
implies cosystolic expansion \emph{over any group}.
