  We study the exponentiation of elements of the gauge Lie algebras ${\rm
hs}(\lambda)$ of three-dimensional higher spin theories. Exponentiable elements
generate one-parameter groups of finite higher spin symmetries. We show that
elements of ${\rm hs}(\lambda)$ in a dense set are exponentiable, when pictured
in certain representations of ${\rm hs}(\lambda)$, induced from representations
of $SL(2,\mathbb{R})$ in the complementary series. We also provide a geometric
picture of higher spin gauge transformations clarifying the physical origin of
these representations. This allows us to construct an infinite-dimensional
topological group $HS(\lambda)$ of finite higher spin symmetries.
Interestingly, this construction is possible only for $0 \leq \lambda \leq 1$,
which are the values for which the higher spin theory is believed to be unitary
and for which the Gaberdiel-Gopakumar duality holds. We exponentiate explicitly
various commutative subalgebras of ${\rm hs}(\lambda)$. Among those, we
identify families of elements of ${\rm hs}(\lambda)$ exponentiating to the unit
of $HS(\lambda)$, generalizing the logarithms of the holonomies of BTZ black
hole connections. Our techniques are generalizable to the Lie algebras relevant
to higher spin theories in dimensions above three.
