Baumslag Solitar Groups Research Articles

Conjugacy separability of HNN-extensions of abelian groups

!. Introduction. A group G is said to be conjugacy separable if, for each pair of nonconjugate elements x, y ~ G, there exists a finite homomorphic image G of G such that the images of x, y in G are not conjugate in G. Conjugacy separability of groups is related to the conjugacy problem in the study of groups. In fact, Mostowski [7] proved that finitely presented conjugacy separable groups have solvable conjugacy problem. Since Stebe [9] and Dyer [2] studied the conjugacy separability of generalized free products of free or nilpotent groups, the conjugacy separability of generalized free products of various groups has been studied in [3, 4, 5, 8, I0, 11]. But the conjugacy separability of HNNextensions was not much known, because one of the simplest type of HNN-extensions, the Baumslag-Solitar group, (b, t : t l b Z t = b 3) is not even residually finite. However Andreadakis, Raptis and Varsos [1] characterized the residual finiteness of HNN-extensions of abelian groups. In [5] Kim, McCarron and Tang characterized the conjugacy separability of l-relator groups of the form (b, t : t -~bPP= b~). Hence the conjugacy separability of HNN-extensions of cyclic groups is known (Theorem 2.6). In [6] Kim and Tang gave necessary and sufficient conditions for HNN-extensions of abelian groups with cyclic associated subgroups to be residually finite and ~ . The purpose of this paper is to study the conjugacy separability of those HNN-extensions, We show that if either the associated subgroups intersect trivially or the associated subgroups have a common subgroup of the same index then the HNN-extensions are conjugacy separable. Applying this result we completely characterize the coniugacy separability of HNN-extensions of abelian groups with cyclic associated subgroups (Theorem 3,6).

CSA-groups and separated free constructions

A group G is called a CSA-group if all its maximal Abelian subgroups are malnormal; that is, Mx ∩ M = 1 for every maximal Abelian subgroup M and x ∈ G − M. The class of CSA-groups contains all torsion-free hyperbolic groups and all groups freely acting on λ-trees. We describe conditions under which HNN-extensions and amalgamated products of CSA-groups are again CSA. One-relator CSA-groups are characterised as follows: a torsion-free one-relator group is CSA if and only if it does not contain F2 × Z or one of the nonabelian metabelian Baumslag-Solitar groups B1, n = 〈x, y | yxy−1 = xn〉, n ∈ Z ∂ {0, 1}; a one-relator group with torsion is CSA if and only if it does not contain the infinite dihedral group.

Growth Functions for Some Nonautomatic Baumslag-Solitar Groups

The growth function of a group is a generating function whose coefficients ${a_n}$ are the number of elements in the group whose minimum length as a word in the generators is n. In this paper we use finite state automata to investigate the growth function for the Baumslag-Solitar group of the form $\langle a,b|{a^{ - 1}}ba = {a^2}\rangle$ based on an analysis of its combinatorial and geometric structure. In particular, we obtain a set of length-minimal normal forms for the group which, although it does not form the language of a finite state automata, is nevertheless built up in a sufficiently coherent way that the growth function can be shown to be rational. The rationality of the growth function of this group is particularly interesting as it is known not to be synchronously automatic. The results in this paper generalize to the groups $\langle a,b|{a^{ - 1}}ba = {a^m}\rangle$ for all positive integers m.

Growth functions for some nonautomatic Baumslag-Solitar groups

The growth function of a group is a generating function whose coefficients a n {a_n} are the number of elements in the group whose minimum length as a word in the generators is n. In this paper we use finite state automata to investigate the growth function for the Baumslag-Solitar group of the form ⟨ a , b | a − 1 b a = a 2 ⟩ \langle a,b|{a^{ - 1}}ba = {a^2}\rangle based on an analysis of its combinatorial and geometric structure. In particular, we obtain a set of length-minimal normal forms for the group which, although it does not form the language of a finite state automata, is nevertheless built up in a sufficiently coherent way that the growth function can be shown to be rational. The rationality of the growth function of this group is particularly interesting as it is known not to be synchronously automatic. The results in this paper generalize to the groups ⟨ a , b | a − 1 b a = a m ⟩ \langle a,b|{a^{ - 1}}ba = {a^m}\rangle for all positive integers m.

The growth of certain amalgamated free products and HNN-extensions

AbstractHere we mean growth in the sense of Milnor and Gromov. After a brief survey of known results, we compute the growth series of the groups, with respect to generators {x, y}. This is done using minimal normal forms obtained by informal use of judiciously chosen rewrite rules. In both of these examples the growth series is a rational function, and we suspect that this is not the case for the Baumslag-Solitar group.

Baumslag-Solitar groups and some other groups of cohomological dimension two

Baumslag-Solitar groups and some other groups of cohomological dimension two