Abstract
AbstractWe generalize results of Thomas, Allcock, Thom–Petersen, and Kar–Niblo to the first $\ell ^{2}$-Betti number of quotients of certain groups acting on trees by subgroups with free actions on the edge sets of the graphs.
Highlights
The 2-Betti numbers were introduced by Atiyah as dimensions of heat kernels of certain operators on Riemannian manifolds
Technical results about 2-Betti numbers that we need can be found in chapters 6 and 8 of loc. cit
G is C∗-simple if and only if it has trivial amenable radical. These two corollaries highlight the relation between the first 2-Betti number and other areas of geometric group theory
Summary
The 2-Betti numbers were introduced by Atiyah as dimensions of heat kernels of certain operators on Riemannian manifolds. G is C∗-simple if and only if it has trivial amenable radical These two corollaries highlight the relation between the first 2-Betti number and other areas of geometric group theory. This generalizes a result of Luck [7], which covers the case of an amalgamated free product, and a result of Tsouvalas [17, Corollary 3.7].
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