= Solution
With $J=\operatorname{diag}(1,-1)$ these matrices obey $M^\dagger JM=J$ and $\det M=1$, identifying the group as <SU(1,1)>. Products preserve the displayed conjugate-entry pattern and <determinant>; the identity belongs, and
$$
M^{-1}=\begin{pmatrix}\alpha^*&-\beta\\-\beta^*&\alpha\end{pmatrix}
$$
also belongs. Multiplication and inversion are smooth. The real constraint $|\alpha|^2-|\beta|^2=1$ has nonzero gradient on its level set, so the matrices form a smooth real three-dimensional manifold. Associativity is inherited from <matrix multiplication>. These facts establish a <Lie group>, which is noncompact because $|\beta|$ is unbounded.
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