The average order of a permutation.
The generalized commutativity degree in a finite group.
The probability that elements of a rank free group generate a rank subgroup.
The triangle and the open triangle
We show that for percolation on any transitive graph, the triangle condition implies the open triangle condition.
Transience of algebraic varieties in linear groups - applications to generic Zariski density
We study the transience of algebraic varieties in linear groups. In particular, we show that a “non elementary” random walk in escapes exponentially fast from every proper algebraic subvariety. We also treat the case where the random walk takes place in the real points of a semisimple split algebraic group and show such a result for a wide family of random walks.As an application, we prove that generic subgroups (in some sense) of linear groups are Zariski dense.