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Finite higher spin transformations from exponentiation

We study the exponentiation of elements of the gauge Lie algebras ${\rm hs}(λ)$ of three-dimensional higher spin theories. Exponentiable elements generate one-parameter groups of finite higher spin symmetries. We show that elements of ${\rm hs}(λ)$ in a dense set are exponentiable, when pictured in certain representations of ${\rm hs}(λ)$, 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 $HS(λ)$ of finite higher spin symmetries. Interestingly, this construction is possible only for $0 \leq λ\leq 1$, 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 ${\rm hs}(λ)$. Among those, we identify families of elements of ${\rm hs}(λ)$ exponentiating to the unit of $HS(λ)$, 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.