Bond percolations on Cayley graphs are probabilistic processes in which edges are deleted randomly in an invariant way. In this talk, we will consider the question about the existence of a unique infinite cluster, which has deep links to geometry, group theory and ergodic theory. We first show a probabilistic characterization of Kazhdan's property (T) which roughly speaking asserts that dense bond percolations form a unique infinite cluster. A complimentary result that Kazhdan groups admit some sparse bond percolation with a unique infinite cluster was proved in a recent breakthrough of Hutchcroft and Pete as the central ingredient in establishing that these groups have cost $1$. Motivated by the fixed price conjecture, the question whether such models can be built as a factor of i.i.d. was posed implicitly there and explicitly by Pete and Rokob. We show that the answer is affirmative for co-compact lattices in connected higher rank semisimple real Lie groups with property (T). Our proof uses a new phenomenon in continuum percolation on the associated symmetric space, which builds on recent breakthrough results about Poisson--Voronoi tessellations by Fraczyk, Mellick and Wilkens.
 
Based on joint works with Chiranjib Mukherjee (Munster) and with Jan Grebik (Leipzig).