Research Projects

Researchers propose to study locally compact groups admitting recurrent random walks, that is, locally compact groups having probability measures μ with the property that ∑ μ ^n is not a radon measure. In this direction a long standing conjecture due to Guivarch and Keane states that a locally compact group admits a recurrent random walk if and only if it has at most quadratic growth, that is, {m(K^n)÷n^2} is bounded for any compact neighbourhood $K$ where $m$ is the Haar measure on $G$. We have proved the conjecture for linear groups - a joint work with Y. Guivarch - and hence we would like to explore the conjecture further by use of finite-dimensional representations of groups.