We applied the framework of maximal invariant to study invariant divergences in statistics:

- The chi-squared divergence is a maximal invariant for the group action of
linear fractional transformations of SL(2,R) on Cauchy distributions.
Since the chi-squared divergence between Cauchy distributions is symmetric,
we deduced that all f-divergences between Cauchy distributions are symmetric and
expressed as a scalar function of their chi-squared divergence.

See On f-Divergences Between Cauchy Distributions (IEEE Trans Information Theory, 2023) also on arxiv:2101.12459

- The triplet of pairwise Minkowski inner products of parameters of hyperboloid distributions
is a maximal invariant for the action of group SO0(1,2).
Since f-divergences between hyperboloid distributions are also invariant by the action,
we deduce that all f-divergences between hyperboloid distributions are functions of 3 canonical terms.

See On the f-divergences between hyperboloid and Poincaré distributions (GSI'23, technical report: 2205.13984)

- The determinant and trace involving the 2x2 SPD matrix parameter
of Poincaré distributions is a maximal invariant for the congruence action of SL(2,R).
Since f-divergences between Poincaré distributions are also invariant under the action, we deduce that all f-divergences
between Poincaré distributions are expressed using the 3 canonical terms.

See On the f-divergences between hyperboloid and Poincaré distributions (GSI'23, technical report: 2205.13984)