/sci/ - Science & Math

On Von Neumann's wiki page, it's stated that he argued for a group-theoretic interpretation of the fact that the "problem of measure" on [math]\mathbb{R}^n[/math] admits a positive solution for [math]n=1,2[/math] and a negative one (due to Banach-Tarski) for [math]n \ge 3[/math]: "the existence of a measure could be determined by looking at the properties of the transformation group of the given space [...] this comes from the fact that the Euclidean group is solvable for [math]n \le 2[/math], and unsolvable for [math]n \ge 3[/math]".

QUESTION: It's known that a random walk on [math]\mathbb{Z}^d[/math] is recurrent for [math]d=1,2[/math] and transient for [math]d \ge 3[/math]. Can /mg/ offer a similar group-theoretic perspective on this?