We show the non-triviality of the mod 3 Chern class of degree 324 of the adjoint representation of the exceptional Lie group $E_{8}$.

We give a volume formula of hyperbolic knot complements using twisted Alexander invariants.

A complex hyperbolic triangle group is a group generated by three complex involutions fixing complex lines in complex hyperbolic space. In our previous papers~[4,5,6,7,8] we discussed complex hyperbolic triangle groups. In particular, in~[5,8] we considered complex hyperbolic triangle groups of type $(n,n,\infty;k)$ and proved that for $n \geq 22$ these groups are not discrete. In this paper we show that if $n \geq 14$, then complex hyperbolic triangle groups of type $(n,n,\infty;k)$ are not discrete and give a new list of non-discrete groups of type $(n,n,\infty;k)$.