A connection between Markov chains on finite simple semigroups and fundamental groups

Let $(S,\circ)$ be a finite simple group, $s_i$, $i=1,\dots,n$, be fixed (not necessarily distinct) elements of $S$, and let $E_{\alpha_1},E_{\alpha_2},\dots, E_{\alpha_{k+1}}$ be a random realisation of a chain of states of a simple homogeneous irreducible Markov chain with the set of states $\{E_1,E_2,\dots,E_n\}$. We study convergence conditions and limit distributions for the sequences of random products of the form $\eta^{(k)}=s_{\alpha_1} \circ s_{\alpha_2}\circ \ldots \circ s_{\alpha_{k+1}}$. The convergence conditions are formulated in terms of some homomorphism from the fundamental group of the transition graph of the Markov chain to the structural group of the semigroup $S$.
This research was supported by the program of the President of the Russian Federation for support of leading scientific schools, grant 8564.2006.10.