110 research outputs found
On a characterization of azumaya algebras
A direct proof of Braun's characterization of Azumaya algebras is given
On the intersection of free subgroups in free products of groups
Let (G_i | i in I) be a family of groups, let F be a free group, and let G =
F *(*I G_i), the free product of F and all the G_i. Let FF denote the set of
all finitely generated subgroups H of G which have the property that, for each
g in G and each i in I, H \cap G_i^{g} = {1}. By the Kurosh Subgroup Theorem,
every element of FF is a free group. For each free group H, the reduced rank of
H is defined as r(H) = max{rank(H) -1, 0} in \naturals \cup {\infty} \subseteq
[0,\infty]. To avoid the vacuous case, we make the additional assumption that
FF contains a non-cyclic group, and we define sigma := sup{r(H\cap
K)/(r(H)r(K)) : H, K in FF and r(H)r(K) \ne 0}, sigma in [1,\infty]. We are
interested in precise bounds for sigma. In the special case where I is empty,
Hanna Neumann proved that sigma in [1,2], and conjectured that sigma = 1;
almost fifty years later, this interval has not been reduced. With the
understanding that \infty/(\infty -2) = 1, we define theta := max{|L|/(|L|-2) :
L is a subgroup of G and |L| > 2}, theta in [1,3]. Generalizing Hanna Neumann's
theorem, we prove that sigma in [theta, 2 theta], and, moreover, sigma = 2
theta if G has 2-torsion. Since sigma is finite, FF is closed under finite
intersections. Generalizing Hanna Neumann's conjecture, we conjecture that
sigma = theta whenever G does not have 2-torsion.Comment: 28 pages, no figure
On the local-indicability cohenâlyndon theorem
For a group H and a subset X of H, we let HX denote the set {hxh?1 | h ? H, x ? X}, and when X is a free-generating set of H, we say that the set HX is a Whitehead subset of H. For a group F and an element r of F, we say that r is CohenâLyndon aspherical in F if F{r} is a Whitehead subset of the subgroup of F that is generated by F{r}. In 1963, Cohen and Lyndon (D. E. Cohen and R. C. Lyndon, Free bases for normal subgroups of free groups, Trans. Amer. Math. Soc. 108 (1963), 526â537) independently showed that in each free group each non-trivial element is CohenâLyndon aspherical. Their proof used the celebrated induction method devised by Magnus in 1930 to study one-relator groups. In 1987, Edjvet and Howie (M. Edjvet and J. Howie, A CohenâLyndon theorem for free products of locally indicable groups, J. Pure Appl. Algebra 45 (1987), 41â44) showed that if A and B are locally indicable groups, then each cyclically reduced element of A*B that does not lie in A ? B is CohenâLyndon aspherical in A*B. Their proof used the original CohenâLyndon theorem. Using BassâSerre theory, the original CohenâLyndon theorem and the EdjvetâHowie theorem, one can deduce the local-indicability CohenâLyndon theorem: if F is a locally indicable group and T is an F-tree with trivial edge stabilisers, then each element of F that fixes no vertex of T is CohenâLyndon aspherical in F. Conversely, by BassâSerre theory, the original CohenâLyndon theorem and the EdjvetâHowie theorem are immediate consequences of the local-indicability CohenâLyndon theorem. In this paper we give a detailed review of a BassâSerre theoretical form of Howie induction and arrange the arguments of Edjvet and Howie into a Howie-inductive proof of the local-indicability CohenâLyndon theorem that uses neither Magnus induction nor the original CohenâLyndon theorem. We conclude with a review of some standard applications of CohenâLyndon asphericit
Non-orientable surface-plus-one-relation groups
Recently DicksâLinnell determined the L2-Betti numbers of the orientable surface-plus-one-relation groups, and their arguments involved some results that were obtained topologically by Hempel and Howie. Using algebraic arguments, we now extend all these results of Hempel and Howie to a larger class of two-relator groups, and we then apply the extended results to determine the L2-Betti numbers of the non-orientable surface-plus-one-relation group
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