the invariance of f-divergences (e.g., Kullback-Leibler divergence, chi-squared divergence, etc.) with respect to real linear fractional transformations (group action of SL(2,R)), and show that all f-divergences between univariate Cauchy densities are symmetric. References Peter McCullagh, "Conditional inference and Cauchy models", Biometrika, volume 79 (1992), pages 247–259. PDF from McCullagh's homepage. Frank Nielsen and Kazuki Okamura, "On f-divergences between Cauchy distributions", arXiv 2101.12459 (2021). Continuous distributions