Random Walks in Soluble Group Theory
Peter Kropholler (University of Southampton)
Friday 6th October, 2017 15:00-16:00 Seminar room 311B
A graph has nodes or vertices connected by edges. A random walk is made by choosing a starting point in the graph and then choosing edges at random to walk successively to new nodes. The return probability is the probability function is the probability of returning to the starting point in n steps. We’ll look at graphs that come from groups.
Usually for a group, that probability decays exponentially with n. Kesten proved that the decay is always better than expected for amenable groups. And he proved that the decay is always exponential for non-amenable groups.
Amenable groups were introduced by John von Neumann. He also created the theory of cellular automata, although many years later. We’ll discover a connection between these two apparently unrelated properties. It’s interesting to speculate whether von Neumann would have been surprised by the connection between two ideas that emerged from very different times in his career.
Von Neumann knew that soluble groups are amenable. Soluble groups were discovered by the French mathematician Evariste Galois, who tragically died following a dual in 1832 at the age of just 20 years and 7 months. After Galois’ death, mathematicians realised that his ideas has completely changed the face of pure mathematics.
To this day soluble groups remain a fundamental source of interesting examples. We’ll explore the connections with return probabilities, amenabilty and cellular automata in this talk.