Finite higher spin transformations from exponentiation

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 $\mathrm{h}\mathrm{s}(\lambda )$ in a dense set are exponentiable, when pictured in certain representations of $\mathrm{h}\mathrm{s}(\lambda )$, 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 $0\le \lambda \le 1$, 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 $\mathrm{h}\mathrm{s}(\lambda )$. Among those, we identify families of elements of $\mathrm{h}\mathrm{s}(\lambda )$ 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.