A quantitative Neumann lemma for finitely generated groups (Q6635463)

The paper under review deals with some classical topics of geometric group theory and combinatorial group theory, namely with the covering theory. This theory has largely influenced many areas of pure mathematics such as topology and algebra. Kontorovich, Kegel, Suzuki and Scorza Dragoni gave several contributions so that there exists a well developed covering theory for finite groups and for infinite groups. In particular, the authors focus on a classical result of \textit{B. H. Neumann} [J. Lond. Math. Soc. 29, 236--248 (1954; Zbl 0055.01604)] who proved what is known today as the ``Neumann Lemma'', that is, an infinite discrete finitely generated group \(G\) cannot be covered by a finite number of cosets \(C=Hg\) of infinite index (here \(g\in G\) and \(H\) is a subgroup of \(G\)). In fact Bernhard Neumann gave the first characterization of the center-by-finite groups (i.e.: groups with center of finite index) by the presence of finite coverings consisting of abelian subgroups, developing from this lemma (along with \textit{R. Baer} [Duke Math. J. 15, 1021--1032 (1948; Zbl 0031.19705)]) the theory of the \(FC\)-groups, that is, the groups with finite conjugacy classes. In other words, controlling the presence of finite coverings of abelian groups one can say how far a group is from being abelian. \N\NThe authors of the present paper show that the original perspective of Bernhard Neumann is still powerful when we ask a quantitative version of the Neumann Lemma, that is, how many cosets of infinite index are needed to cover \(B_r(G, S)\), where \(B_r(G, S)\) denotes the ball of radius \(r\) in \(G\) with \(S\) finite symmetric set of generators of \(G\) ? Working on this problem with the methods of the geometric group theory means that we want to control the size of the so called ``coset covering function'' \(\mathfrak{C}_{G,S}(r)\), that is, the minimum positive integer \(N\) for which there exist cosets of infinite index \(C_1, C_2, \ldots, C_N\) such that their union covers \(B_r(G, S)\). Theorems 1, 2 and 3 (of the paper under review) show some natural restrictions for \(\mathfrak{C}_{G,S}(r)\), discussing their meaning in connection with the propery \((T)\) of Kazhdan and with the theory of the random walks.