ROC represents and leads an Orthodox branch of Christianity.
AFAIK originally Byzantine (Eastern Roman Empire) was a centre of Orthodoxy before it’s capital Constantinople (now Istanbul)was took over by Turks in 1453. If I’m not mistaken a world biggest mosque in Istanbul is actually a remodeled Orthodox cathedral.
After Turks took over Byzantine ROC or Moscow patriarchate took lead over Orthodox branch. It has influenced Russian politics at Balkans and Central Asia for centuries.
For now Moscow and Patriarch Kirill for orthodoxes is like a Vatican and Pope for catholics and this desecreted cathedral is their main base.
I’m not a Ph.D in history. Correct me if I’m wrong.
Hagia Sofia would be better described as a desecrated Christian Cathedral. It dates to AD360, when Eastern and Western Christendom were still fully in communion with each other. It has been a museum run by the officially secular Turkish government since the 1930s.