arXiv subjects
Publications and source records attributed to Warren Dicks.
Revised 1977 seminar handout on Quillen's proof of the 1976 Quillen-Suslin theorem -- that all finitely generated, projective $k[x_1,\ldots,x_n]$-modules are free. This particular write-up is based on arguments of Quillen, Paul Roberts, Vaserstein, and Moshe Roitman -- arguments that are here translated into the language of idempotent matrices, which seems rather natural.
In 1989, Dicks and Dunwoody proved the Almost Stability Theorem, which has among its corollaries the Stallings-Swan theorem that groups of cohomological dimension one are free. In this article, we use a nestedness result of Bergman, Bowditch, and Dunwoody to simplify somewhat the proof of the finitely generable case of the Almost Stability Theorem. We also simplify the proof of the non finitely generable case. The proof we give here of the Almost Stability Theorem is essentially self contained, except that in the non finitely generable case we refer the reader to the original argument for the proofs of two technical lemmas about groups acting on trees.
J.H.C. Whitehead's second free-group algorithm determines whether or not two given elements of a free group lie in the same orbit of the automorphism group of the free group. The algorithm involves certain connected graphs, and Whitehead used three-manifold models to prove their connectedness; later, Rapaport and Higgins & Lyndon gave group-theoretic proofs. Combined work of Gersten, Stallings, and Hoare showed that the three-manifold models may be viewed as graphs. We give the direct translation of Whitehead's topological argument into the language of graph theory.
Let $F$ be any finite-rank free group, and $R$ be any finite subset of $\{g, [g]: g \in F-\{1\}\}$, where $[g]:= \{fgf^{-1}:f\in F\}$. By an $R$-allocating $F$-factorization we mean a set $\mathcal{H}$ of nontrivial subgroups of $F$ such that $\ast_{H \in \mathcal{H}} H = F$ and $R \subseteq \{h, [h] : h \in H, H\in \mathcal{H}\}$. We show that Whitehead's (fast) cutvertex algorithm inputs the pair $(F,R)$ and outputs a maximum-size $R$-allocating $F$-factorization. Richard Stong showed this in the case where $R \subseteq F$ or $R \subseteq \{[g] : g \in F\}$, thereby unifying and generalizing a collection of results obtained by Berge, Bestvina, Lyon, Shenitzer, Stallings, Starr, and Whitehead. Our proof is based on the interaction between two normal forms for the elements of $F$, rather than the algebraic topology of handlebodies, trees, or graph folding.
We draw attention to an easy-to-remember explanation for the graded-case inequality of Golod and Shafarevich. We review some of the classic material on this inequality.
Let G be a group and H be a subgroup of G. We say that H is left relatively convex in G if the left G-set G/H has at least one G-invariant order; when G is left orderable, this holds if and only if H is convex in G under some left ordering of G. We give a criterion for H to be left relatively convex in G that generalizes a famous theorem of Burns and Hale and has essentially the same proof. We show that all maximal cyclic subgroups are left relatively convex in free groups, in right-angled Artin groups, and in surface groups that are not the Klein-bottle group. The free-group case extends a result of Duncan and Howie. We show that if G is left orderable, then each free factor of G is left relatively convex in G. More generally, for any graph of groups, if each edge group is left relatively convex in each of its vertex groups, then each vertex group is left relatively convex in the fundamental group; this generalizes a result of Chiswell. We show that all maximal cyclic subgroups in locally residually torsion-free nilpotent groups are left relatively convex.
We construct total orders on the vertex set of an oriented tree. The orders are based only on up-down counts at the interior vertices and the edges along the unique geodesic from a given vertex to another. As an application, we provide a short proof (modulo Bass-Serre theory) of Vinogradov's result that the free product of left-orderable groups is left-orderable.
Let $F$ be a finite-rank free group and $H$ be a finite-rank subgroup of $F$. We discuss proofs of two algorithms that sandwich $H$ between an upper-layer free-product factor of $F$ that contains $H$ and a lower-layer free-product factor of $F$ that is contained in $H$. Richard Stong showed that the unique smallest-possible upper layer, denoted $\operatorname{Cl}(H)$, is visible in the output of the polynomial-time cut-vertex algorithm of J. H. C. Whitehead. Stong's proof used bi-infinite paths in a Cayley tree and sub-surfaces of a three-manifold. We give a variant of his proof that uses edge-cuts of the Cayley tree induced by edge-cuts of a Bass-Serre tree. A. Clifford and R. Z. Goldstein gave an exponential-time algorithm that determines whether or not the trivial subgroup is the only possible lower layer. Their proof used Whitehead's three-manifold techniques. We give a variant of their proof that uses Whitehead's cut-vertex results, and thereby obtain a somewhat simpler algorithm that yields a lower layer of maximum-possible rank.
Let $R$ be a ring (associative, with 1), and let $R<< a,b>>$ denote the power-series $R$-ring in two non-commuting, $R$-centralizing variables, $a$ and $b$. Let $A$ be an $R$-subring of $R<< a>>$ and $B$ be an $R$-subring of $R<< b>>$, and let $\alpha$ denote the natural map $A \amalg_R B \to R<< a,b>>$. This article describes some situations where $\alpha$ is injective and some where it is not. We prove that if $A$ is a right Ore localization of $R[a]$ and $B$ is a right Ore localization of $R[b]$, then $\alpha$ is injective. For example, the group ring over $R$ of the free group on $\{1+a, 1+b\}$ is $R[ (1+a)^{\pm 1}] \amalg_R R[ (1+b)^{\pm 1}]$, which then embeds in $R<< a,b>>$. We thus recover a celebrated result of R H Fox, via a proof simpler than those previously known. We show that $\alpha$ is injective if $R$ is \textit{$\Pi$-semihereditary}, that is, every finitely generated, torsionless, right $R$-module is projective. The article concludes with some results contributed by G M Bergman that describe situations where $\alpha$ is not injective. He shows that if $R$ is commutative and $\text{w.gl.dim\,} R \ge 2$, then there exist examples where the map $\alpha' \colon A \amalg_R B \to R<< a>>\amalg_R R<< b>>$ is not injective, and hence neither is $\alpha$. It follows from a result of K R Goodearl that when $R$ is a commutative, countable, non-self-injective, von Neumann regular ring, the map $\alpha"\colon R<< a>>\amalg_R R<< b>> \to R<< a,b>>$ is not injective. Bergman gives procedures for constructing other examples where $\alpha"$ is not injective.
For a connected graph L, let G(L) be a group with generators the vertex set of L, subject only to the relations that the ends of each edge commute. Now let H(L) be the kernel of the homomorphism from G(L) to the integers that takes each vertex to 1. M. Bestvina and N. Brady have shown that finiteness properties of H(L) are intimately related to the topology of the clique complex of L. We give a presentation for H(L), with generators the edges of L, and an infinite family of relators for each 1-cycle in L. In the case when the clique complex for L is simply-connected, we give a finite presentation for H(L), with generators the edges (or 2-cliques) of L, and two relators for each 3-clique in L.
Let $m, m', r, r',t, t'$ be positive integers with $r, r' \ge 2$. Let $L_r$ denote the ring that is universal with an invertible $1 \times r$ matrix. Let $M_m(L_r^{\otimes t})$ denote the ring of $m \times m$ matrices over the tensor product of $t$ copies of $L_r$. In a natural way, $M_m(L_r^{\otimes t})$ is a partially ordered ring with involution. Let $PU_m(L_r^{\otimes t})$ denote the group of positive unitary elements. We show that $PU_m(L_r^{\otimes t})$ is isomorphic to the Brin-Higman-Thompson group $t V_{r,m}$; the case $t =1$ was found by Pardo, that is, $PU_m(L_r)$ is isomorphic to the Higman-Thompson group $V_{r,m}$. We survey arguments of Abrams, \'Anh, Bleak, Brin, Higman, Lanoue, Pardo, and Thompson that prove that $t' V_{r',m'} \cong tV_{r,m} $ if and only if $r' = r$, $t'=t$ and $ \gcd(m',r'-1) = \gcd(m,r-1)$ (if and only if $M_{m'}(L_{r'}^{\otimes t'})$ and $M_m(L_r^{\otimes t})$ are isomorphic as partially ordered rings with involution).
Let g and p be non-negative integers. Let A(g,p) denote the group consisting of all those automorphisms of the free group on {t_1,...,t_p, x_1,...,x_g, y_1,...y_g} which fix the element t_1t_2...t_p[x_1,y_1]...[x_g,y_g] and permute the set of conjugacy classes {[t_1],....,[t_p]}. Labru\`ere and Paris, building on work of Artin, Magnus, Dehn, Nielsen, Lickorish, Zieschang, Birman, Humphries, and others, showed that A(g,p) is generated by a set that is called the ADLH set. We use methods of Zieschang and McCool to give a self-contained, algebraic proof of this result. Labru\`ere and Paris also gave defining relations for the ADLH set in A(g,p); we do not know an algebraic proof of this for g > 1. Consider an orientable surface S(g,p) of genus g with p punctures, such that (g,p) is not (0,0) or (0,1). The algebraic mapping-class group of S(g,p), denoted M(g,p), is defined as the group of all those outer automorphisms of the one-relator group with generating set {t_1,...,t_p, x_1,...,x_g, y_1,...y_g} and relator t_1t_2...t_p[x_1,y_1]...[x_g,y_g] which permute the set of conjugacy classes {[t_1],....,[t_p]}. It now follows from a result of Nielsen that M(g,p) is generated by the image of the ADLH set together with a reflection. This gives a new way of seeing that M(g,p) equals the (topological) mapping-class group of S(g,p), along lines suggested by Magnus, Karrass, and Solitar in 1966.
For a group $H$ and a subset $X$ of $H$, we let ${}^HX$ denote the set $\{hxh^{-1} \mid h \in H, x \in X\}$, 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 Cohen-Lyndon aspherical in $F$ if ${}^F\{r\}$ is a Whitehead subset of the subgroup of $F$ that is generated by ${}^F\{r\}$. In 1963, D. E. Cohen and R. C. Lyndon independently showed that in each free group each non-trivial element is Cohen-Lyndon aspherical. In 1987, M. Edjvet and J. Howie showed that if $A$ and $B$ are locally indicable groups, then each cyclically reduced element of $A \ast B$ that does not lie in $A \cup B$ is Cohen-Lyndon aspherical in $A \ast B$. Using Bass-Serre Theory and the Edjvet-Howie Theorem, one can deduce the local-indicability Cohen-Lyndon Theorem: if $F$ is a locally indicable group and $T$ is an $F$-tree with trivial edge stabilizers, then each element of $F$ that fixes no vertex of $T$ is Cohen-Lyndon aspherical in $F$. Conversely, the Cohen-Lyndon Theorem and the Edjvet-Howie Theorem are immediate consequences of the local-indicability Cohen-Lyndon Theorem. In this article, we give a detailed review of Howie induction and arrange the arguments of Edjvet and Howie into a Howie-inductive proof of the local-indicability Cohen-Lyndon Theorem that does not use Magnus induction or the Cohen-Lyndon Theorem. We conclude with a review of some standard applications of Cohen-Lyndon asphericity.
To each once-punctured-torus bundle, $T_\phi$, over the circle with pseudo-Anosov monodromy $\phi$, there are associated two tessellations of the complex plane: one, $\Delta(\phi)$, is (the projection from $\infty$ of) the triangulation of a horosphere at $\infty$ induced by the canonical decomposition into ideal tetrahedra, and the other, $CW(\phi)$, is a fractal tessellation given by the Cannon-Thurston map of the fiber group switching back and forth between gray and white each time it passes through $\infty$. In this paper, we study the relation between $\Delta(\phi)$ and $CW(\phi)$.
Recently Dicks-Linnell determined the $L^2$-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 $L^2$-Betti numbers of the non-orientable surface-plus-one-relation groups.
A right-angled Coxeter group is a group with a given set of generators of order two, subject only to the relations that certain pairs of the generators commute. Various papers have shown how homological properties of the Coxeter group are related to homological properties of the simplicial complex whose simplices are the sets of commuting generators. Using these techniques, we construct torsion-free groups which are Poincare duality groups over some rings but not over others, and a group whose integral cohomological dimension is finite but strictly greater than its cohomological dimension over any field. We determine which Coxeter groups have finite virtual cohomological dimension (it is classical that all finitely generated Coxeter groups have finite vcd, but there are others). We also give minimal presentations for certain torsion-free finite-index subgroups of right-angled Coxter groups. Finally we give a `bare-hands' construction (using free products with amalgamation and HNN extensions) of a torsion-free group whose integral cohomological dimension is strictly greater than its rational cohomological dimension.
This article surveys many standard results about the braid group with emphasis on simplifying the usual algebraic proofs. We use van der Waerden's trick to illuminate the Artin-Magnus proof of the classic presentation of the algebraic mapping-class group of a punctured disc. We give a simple, new proof of the Dehornoy-Larue braid-group trichotomy, and, hence, recover the Dehornoy right-ordering of the braid group. We then turn to the Birman-Hilden theorem concerning braid-group actions on free products of cyclic groups, and the consequences derived by Perron-Vannier, and the connections with the Wada representations. We recall the very simple Crisp-Paris proof of the Birman-Hilden theorem that uses the Larue-Shpilrain technique. Studying ends of free groups permits a deeper understanding of the braid group; this gives us a generalization of the Birman-Hilden theorem. Studying Jordan curves in the punctured disc permits a still deeper understanding of the braid group; this gave Larue, in his PhD thesis, correspondingly deeper results, and, in an appendix, we recall the essence of Larue's thesis, giving simpler combinatorial proofs.
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.