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Register a new userVirtual rational Betti numbers of nilpotent-by-abelian groups
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In this paper we study virtual rational Betti numbers of a nilpotent-by-abelian group $G$, where the abelianization $N/N$ of its nilpotent part $N$ satisfies certain tameness property. More precisely, we prove that if $N/N$ is $2(c(n-1)-1)$-tame as a $G/N$-module, $c$ the nilpotency class of $N$, then $mathrm{vb}_j(G):=sup_{Minmathcal{A}_G}dim_mathbb{Q} H_j(M,mathbb{Q})$ is finite for all $0leq jleq n$, where $mathcal{A}_G$ is the set of all finite index subgroups of $G$.
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Refining a result of Erdos and Mays, we give asymptotic series expansions for the functions $A(x)-C(x)$, the count of $nleq x$ for which every group of order $n$ is abelian (but not all cyclic), and $N(x)-A(x)$, the count of $nleq x$ for which every group of order $n$ is nilpotent (but not all abelian).
We apply a construction developed in a previous paper by the authors in order to obtain a formula which enables us to compute $ell^2$-Betti numbers coming from a family of group algebras representable as crossed product algebras. As an application, we obtain a whole family of irrational $ell^2$-Betti numbers arising from the lamplighter group algebra $K[mathbb{Z}_2 wr mathbb{Z}]$, being $K$ a subfield of the complex numbers closed under complex conjugation. This procedure is constructive, in the sense that one has an explicit description of the elements realizing such irrational numbers. This extends the work made by Grabowski, who first computed irrational $ell^2$-Betti numbers from the algebras $mathbb{Q}[mathbb{Z}_n wr mathbb{Z}]$, where $n geq 2$ is a natural number. We also apply the techniques developed to the (generalized) odometer algebra $mathcal{O}(overline{n})$, where $overline{n}$ is a supernatural number. We compute its $*$-regular closure, and this allows us to fully characterize the set of $ell^2$-Betti numbers arising from $mathcal{O}(overline{n})$.
Full residual finiteness growth of a finitely generated group $G$ measures how efficiently word metric $n$-balls of $G$ inject into finite quotients of $G$. We initiate a study of this growth over the class of nilpotent groups. When the last term of the lower central series of $G$ has finite index in the center of $G$ we show that the growth is precisely $n^b$, where $b$ is the product of the nilpotency class and dimension of $G$. In the general case, we give a method for finding an upper bound of the form $n^b$ where $b$ is a natural number determined by what we call a terraced filtration of $G$. Finally, we characterize nilpotent groups for which the word growth and full residual finiteness growth coincide.
Let $sigma ={sigma_{i} | iin I}$ be a partition of the set $Bbb{P}$ of all primes and $G$ a finite group. A chief factor $H/K$ of $G$ is said to be $sigma$-central if the semidirect product $(H/K)rtimes (G/C_{G}(H/K))$ is a $sigma_{i}$-group for some $i=i(H/K)$. $G$ is called $sigma$-nilpotent if every chief factor of $G$ is $sigma$-central. We say that $G$ is semi-${sigma}$-nilpotent (respectively weakly semi-${sigma}$-nilpotent) if the normalizer $N_{G}(A)$ of every non-normal (respectively every non-subnormal) $sigma$-nilpotent subgroup $A$ of $G$ is $sigma$-nilpotent. In this paper we determine the structure of finite semi-${sigma}$-nilpotent and weakly semi-${sigma}$-nilpotent groups.
The Tarski number of a non-amenable group G is the minimal number of pieces in a paradoxical decomposition of G. In this paper we investigate how Tarski numbers may change under various group-theoretic operations. Using these estimates and known properties of Golod-Shafarevich groups, we show that the Tarski numbers of 2-generated non-amenable groups can be arbitrarily large. We also use the cost of group actions to show that there exist groups with Tarski numbers 5 and 6. These provide the first examples of non-amenable groups without free subgroups whose Tarski number has been computed precisely.
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