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Register a new userNilpotent completions of groups, Grothendieck pairs, and four problems of Baumslag
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Two groups are said to have the same nilpotent genus if they have the same nilpotent quotients. We answer four questions of Baumslag concerning nilpotent completions. (i) There exists a pair of finitely generated, residually torsion-free-nilpotent groups of the same nilpotent genus such that one is finitely presented and the other is not. (ii) There exists a pair of finitely presented, residually torsion-free-nilpotent groups of the same nilpotent genus such that one has a solvable conjugacy problem and the other does not. (iii) There exists a pair of finitely generated, residually torsion-free-nilpotent groups of the same nilpotent genus such that one has finitely generated second homology $H_2(-,Z)$ and the other does not. (iv) A non-trivial normal subgroup of infinite index in a finitely generated parafree group cannot be finitely generated. In proving this last result, we establish that the first $L^2$ betti number of a finitely generated parafree group of rank $r$ is $r-1$. It follows that the reduced $C^*$-algebra of the group is simple if $rge 2$, and that a version of the Freiheitssatz holds for parafree groups.
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In this paper we classify Baumslag-Solitar groups up to commensurability. In order to prove our main result we give a solution to the isomorphism problem for a subclass of Generalised Baumslag-Solitar groups.
We study convergent sequences of Baumslag-Solitar groups in the space of marked groups. We prove that BS(m,n) --> F_2 for |m|,|n| --> infty and BS(1,n) --> Z wr Z for |n| --> infty. For m fixed, |m|>1, we show that the sequence (BS(m,n))_n is not convergent and characterize many convergent subsequences. Moreover if X_m is the set of BS(m,n)s for n relatively prime to m and |n|>1, then the map BS(m,n) mapsto n extends continuously on the closure of X_m to a surjection onto invertible m-adic integers.
Given a $T_0$ paratopological group $G$ and a class $mathcal C$ of continuous homomorphisms of paratopological groups, we define the $mathcal C$-$semicompletion$ $mathcal C[G)$ and $mathcal C$-$completion$ $mathcal C[G]$ of the group $G$ that contain $G$ as a dense subgroup, satisfy the $T_0$-separation axiom and have certain universality properties. For special classes $mathcal C$, we present some necessary and sufficient conditions on $G$ in order that the (semi)completions $mathcal C[G)$ and $mathcal C[G]$ be Hausdorff. Also, we give an example of a Hausdorff paratopological abelian group $G$ whose $mathcal C$-semicompletion $mathcal C[G)$ fails to be a $T_1$-space, where $mathcal C$ is the class of continuous homomorphisms of sequentially compact topological groups to paratopological groups. In particular, the group $G$ contains an $omega$-bounded sequentially compact subgroup $H$ such that $H$ is a topological group but its closure in $G$ fails to be a subgroup.
We prove that for any prime $pgeq 3$ the minimal exponential growth rate of the Baumslag-Solitar group $BS(1,p)$ and the lamplighter group $mathcal{L}_p=(mathbb{Z}/pmathbb{Z})wr mathbb{Z}$ are equal. We also show that for $p=2$ this claim is not true and the growth rate of $BS(1,2)$ is equal to the positive root of $x^3-x^2-2$, whilst the one of the lamplighter group $mathcal{L}_2$ is equal to the golden ratio $(1+sqrt5)/2$. The latter value also serves to show that the lower bound of A.Mann from [Mann, Journal of Algebra 326, no. 1 (2011) 208--217] for the growth rates of non-semidirect HNN extensions is optimal.
We exhibit a weakly aperiodic tile set for Baumslag-Solitar groups, and prove that the domino problem is undecidable on these groups. A consequence of our construction is the existence of an arecursive tile set on Baumslag-Solitar groups.
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