I learnt from a talk that consider a random product of i.i.d. matrices, randomly chosen from SL(2,R): $T\_n=A\_n \cdots A\_2 A\_1$, where the random matrices $A\_i$ are i.i.d.


A classical Furstenberg theorem then implies, that under some very mild nondegeneracy conditions (no finite common invariant set of lines, no common invariant metric) for the law of $A\_i$'s the norm of such a product almost surely grows exponentially.


Now I wonder if this conclusion also holds if the matrices are chosen from SL(2,C)?


For simplicity, I checked the case with two matrices $A=\left[\begin{array}{cccc}i &0\\0 &-i\end{array}\right]$ and $B=\left[\begin{array}{cccc}1+i &-i\\i &1-i\end{array}\right]$. Now I multiply the matrices which are randomly chosen from $A$ and $B$ with probability $1/2$ and $1/2$, and find that indeed the norm of such a product grows exponentially. But I am not sure if there is a proof for this. Thanks!



Is there a generalization of Furstenberg theorem from SL(2,R) to SL(2,C) matrices?
The original work of Furstenberg "Non-commuting random products" (1963) actually contains an answer to your question in Theorem 8.6 which states the positivity of the top Lyapunov exponent for **any** random walk on $SL(d,\mathbb R)$ under the natural first moment condition and the irreducibility condition your mention. So in your case you just have to embed $SL(2,\mathbb C)$ into a higher order real matrix group.


