The Hanna Neumann conjecture states that if F is a free group, then for all nontrivial finitely generated subgroups H,K <= F, rank(H intersect K) - 1 <= [rank(H)-1] [rank(K)-1]. Where most papers to date have considered a direct graph theoretic interpretation of the conjecture, here we consider the use of monomorphisms. We illustrate the effectiveness of this approach with two results. First, we show that for any finitely generated groups H,K <= F either the pair H,K or the pair H^{-}, K satisfy the Hanna Neumann conjecture--Here {-} denotes the automorphism which sends each generator of F to its inverse. Next, using particular monomorphisms from F to F_2, we obtain that if the Hanna Neumann conjecture is false then there is a counterexample H,K < F_2 having the additional property that all the branch vertices in the foldings of H and K are of degree 3, and all degree 3 vertices have the same local structure or ``type''.
We prove the Strengthened Hanna Neumann Conjecture, in its common graph theoretic formulation. Our original approach to this conjecture used cohomology of sheaves on graphs, although here we give a short combinatorial proof that we found in a succession of simplifications of the cohomological approach. Our proof is inductive, and requires Tardos' previous result settling the rank two case of the conjecture.
We give a very short proof that a subgroup of a free group that is positively generated cannot be part of a counterexample to the Generalized Hanna Neumann Conjecture.
We prove the Strengthened Hanna Neumann Conjecture. We give a more direct cohomological interpretation of the conjecture in terms of "typical" covering maps, and use graph Galois theory to "symmetrize" the conjecture. The conjecture is then related to certain kernel of a morphism of sheaves, and is implied provided these kernels are co-acyclic in the covering cohomology theory. This allows us to prove a slightly generalized Strengthened Hanna Neumann Conjecture; this conjecture is false if generalized to all sheaves. The kernels we use do not exist in the theory of graphs, so our use of sheaf …
A new bound for the rank of the intersection of finitely generated subgroups of a free group is given, formulated in topological terms, and very much in the spirit of Stallings. The bound is a contribution to (although unfortunately not a solution of) the strengthened Hanna Neumann conjecture.
In this paper we establish some foundations regarding sheaves of vector spaces on graphs and their invariants, such as homology groups and their limits. We then use these ideas to prove the Hanna Neumann Conjecture of the 1950's; in fact, we prove a strengthened form of the conjecture. We introduce a notion of a sheaf of vector spaces on a graph, and develop the foundations of homology theories for such sheaves. One sheaf invariant, its "maximum excess," has a number of remarkable properties. It has a simple definition, with no reference to homology theory, that resembles graph expansion. Yet …
The Hanna Neumann conjecture states that if F is a free group, then for all finitely generated subgroups H,K <= F, rank(H intersect K) - 1 <= [ rank(H)-1 ] [ rank(K)-1 ] In this paper, we show that if one of the subgroups, say H, has a generating set consisting of only positive words, then H is not part of any counterexample to the conjecture. We further show that if H <= F({a,b}) is part of a counterexample to the conjecture, then its folding Gamma_H must contain source and/or sink vertices.
We confirm the Hanna Neumann conjecture for topologically finitely generated closed subgroups $U$ and $W$ of a nonsolvable Demushkin group $G$. Namely, we show that \begin{equation*} \sum_{g \in U \backslash G/W} \bar d(U \cap gWg^{-1}) \leq \bar d(U) \bar d(W) \end{equation*} where $\bar d(K) = \max\{d(K) - 1, 0\}$ and $d(K)$ is the least cardinality of a topological generating set for the group $K$.
The Hanna Neumann conjecture gives a bound on the intersection of finitely generated subgroups of free groups. We explore a natural extension of this result, which turns out to be true only in the finite index case, and provide counterexamples for the general case. We also see that the graph-based method of generating random subgroups of free groups developed by Bassino, Nicaud and Weil is well-suited to generating subgroups with non-trivial intersections. The same method is then used to generate a counterexample to a similar conjecture of Guzman.
To solve problems of Gilbert Baumslag and Hanna Neumann, posed in the 1960's, we construct a nontrivial variety of groups all of whose noncyclic free groups are non-hopfian.
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 …
We prove a rank 1 version of the Hanna Neumann Theorem. This shows that every one-relator 2-complex without torsion has the nonpositive immersion property. The proof generalizes to staggered and reducible 2-complexes.
In this paper we establish some subnormal embeddings of groups into groups with additional properties; in particular embeddings of countable groups into 2-generated groups with some extra properties. The results obtained are generalizations of theorems of P. Hall, R. Dark, B. Neumann, Hanna Neumann, G. Higman on embeddings of that type. Considering subnormal embeddings of finite groups into finite groups with additional properties we bring some illustration to a result of H. Heineken and J. Lennox.
We construct an efficient model for graphs of finitely generated subgroups of free groups. Using this we give a very short proof of Dicks's reformulation of the strengthened Hanna Neumann Conjecture as the Amalgamated Graph Conjecture. In addition, we answer a question of Culler and Shalen on ranks of intersections in free groups. The latter has also been done independently by R. P. Kent IV.
We prove that the reduced Kurosh rank of the intersection of two subgroups $H$ and $K$ of a free product of right-orderable groups is bounded above by the product of the reduced Kurosh ranks of $H$ and $K$. In particular, taking the fundamental group of a graph of groups with trivial vertex and edge groups, and its Bass-Serre tree, our Theorem becomes the desired inequality of the usual Strengthened Hanna Neumann conjecture for free groups.
We prove an estimate for the rank of the intersection of free subgroups in virtually free groups, which is analogous to the Hanna Neumann inequality for subgroups in a free group and to the S.V. Ivanov estimate for subgroups in free products of groups. We also prove a more general estimate for the rank of the intersection of free subgroups in the fundamental group of a finite graph of groups with finite edge groups.
This note contains a (short) proof of the following generalisation of the Friedman--Mineyev theorem (earlier known as the Hanna Neumann conjecture): if $A$ and $B$ are nontrivial free subgroups of a virtually free group containing a free subgroup of index $n$, then $rank(A\cap B)-1\leqslant n\cdot(rank(A)-1)\cdot(rank(B)-1)$. In addition, we obtain a virtually-free-product analogue of this result.
We determine when a quasi-isometry between discrete spaces is at bounded distance from a bilipschitz map. From this we prove a geometric version of the Von Neumann conjecture on amenability. We also get some examples in geometric groups theory which show that the sign of the Euler characteristic is not a coarse invariant. Finally we get some general results on uniformly finite homology which we will apply to manifolds in a later paper.
We address the problem to characterise closed type I subgroups of the automorphism group of a tree. Even in the well-studied case of Burger-Mozes' universal groups, non-type I criteria were unknown. We prove that a huge class of groups acting properly on trees are not of type I. In the case of Burger-Mozes groups, this yields a complete classification of type I groups among them. Our key novelty is the use of von Neumann algebraic techniques to prove the stronger statement that the group von Neumann algebra of the groups under consideration is non-amenable.
We prove a generalized version of the Strong Atiyah Conjecture for the infinite dihedral group W, replacing the group von Neumann algebra NW with the Hecke-von Neumann algebra N_qW.
Let H and K be subgroups of a free group of ranks h and k \geq h. We prove the following strong form of Burns' inequality: rank(H \cap K) - 1 \leq 2(h-1)(k-1) - (h-1)(rank(H \vee K) -1). A corollary of this, also obtained by L. Louder and D. B. McReynolds, has been used by M. Culler and P. Shalen to obtain information regarding the volumes of hyperbolic 3-manifolds. We also prove the following particular case of the Hanna Neumann Conjecture, which has also been obtained by Louder. If the join of H and K has rank at least (h + k + 1)/2, then the intersection of H and K has rank no more than (h-…
The famous Hanna Neumann Conjecture (now the Friedman-Mineyev theorem) gives an upper bound for the ranks of the intersection of arbitrary subgroups $H$ and $K$ of a non-abelian free group. It is an interesting question to `quantify' this bound with respect to the rank of $H\vee K$, the subgroup generated by $H$ and $K$. We describe a set of realizable values $(rk(H\vee K),rk(H\cap K))$ for arbitrary $H$, $K$, and conjecture that this locus is complete. We study the combinatorial structure of the topological pushout of the core graphs for $H$ and $K$, with the help of graphs introduced by Dick…
Let $Γ$ be a discrete group. Following Linnell and Schick one can define a continuous ring $c(Γ)$ associated with $Γ$. They proved that if the Atiyah Conjecture holds for a torsion-free group $Γ$, then $c(Γ)$ is a skew field. Also, if $Γ$ has torsion and the Strong Atiyah Conjecture holds for $Γ$, then $c(Γ)$ is a matrix ring over a skew field. The simplest example when the Strong Atiyah Conjecture fails is the lamplighter group $Γ=Z_2\wr Z$. It is known that $C(Z_2\wr Z)$ does not even have a classical ring of quotients. Our main result is that if $H$ is amenable, then $c(Z_2\wr H)$ is isomor…
Let G be a group and let K be a field of characteristic zero. We shall prove that KG can be embedded into a von Neumann unit-regular ring. In the course of the proof, we shall obtain a result relevant to the Atiyah conjecture.
Let $G$ be a finite group, $F$ a field, and $V$ a finite dimensional $FG$-module such that $G$ has no trivial composition factor on $V$. Then the arithmetic average dimension of the fixed point spaces of elements of $G$ on $V$ is at most $(1/p) \dim V$ where $p$ is the smallest prime divisor of the order of $G$. This answers and generalizes a 1966 conjecture of Neumann which also appeared in a paper of Neumann and Vaughan-Lee and also as a problem in The Kourovka Notebook posted by Vaughan-Lee. Our result also generalizes a recent theorem of Isaacs, Keller, Meierfrankenfeld, and Moretó. Variou…
We prove that amenability of a discrete group is equivalent to dimension flatness of certain ring inclusions naturally associated with measure preserving actions of the group. This provides a group-measure space theoretic solution to a conjecture of Lück stating that amenability of a group is characterized by dimension flatness of the inclusion of its complex group algebra into the associated von Neumann algebra.
In our previous paper, "l^{p}-Version of von Neumann Dimension for Banach Space Representations of Sofic Groups," we define an extended version of von Neumann dimension for actions of a sofic group on a Banach space. This dimension was studied especially for the translation action of G on l^{p}(G), as well as the multiplication on the non-commutative L^{p} space associated with the group von Neumann algebra. We discuss how one can similarly define an extended dimension for representations of an equivalence relation. We also define an analogue of l^{2}-Betti numbers for equivalence relations in…
We study the Newton polytopes of determinants of square matrices defined over rings of twisted Laurent polynomials. We prove that such Newton polytopes are single polytopes (rather than formal differences of two polytopes); this result can be seen as analogous to the fact that determinants of matrices over commutative Laurent polynomial rings are themselves polynomials, rather than rational functions. We also exhibit a relationship between the Newton polytopes and invertibility of the matrices over Novikov rings, thus establishing a connection with the invariants of Bieri-Neumann-Strebel (BNS)…
Kapovich and Nagnibeda introduced the space $\mathcal{S} {\rm Curr}(F_N)$ of subset currents on a free group $F_N$ of rank $N\geq 2$, which can be thought of as a measure-theoretic completion of the set of all conjugacy classes of finitely generated subgroups of $F_N$. We define a product $\mathcal{N} (H,K)$ of two finitely generated subgroups $H$ and $K$ of $F_N$ by the sum of the reduced rank $\overline{\rm rk}(H\cap gKg^{-1})$ over all double cosets $HgK\ (g\in F_N)$, and extend the product $\mathcal{N}$ to a continuous symmetric $\mathbb{R}_{\geq 0}$-bilinear functional $\mathcal{N} \colon…
Classical Hecke operators on Maass forms are unitarely equivalent, up to a commuting phase, to completely positive maps on II$_1$ factors, associated to a pair of isomorphic subfactors, and an intertwining unitary. This representation is obtained through a quantized representation of the Hecke operators. in this representation, the Hecke operators act on the Berezin's quantization, deformation algebra of the fundamental domain of $\PSL(2,\Z)$ in the upper halfplane. The Hecke operators are inheriting from the ambient, non-commutative algebra on which they act, a rich structure of matrix inequa…
In this article we examine the conjecture of Neumann and Reid that the only hyperbolic knots in the $3$-sphere which admit hidden symmetries are the figure-eight knot and the two dodecahedral knots. Knots whose complements cover hyperbolic reflection orbifolds admit hidden symmetries, and we verify the Neumann-Reid conjecture for knots which cover small hyperbolic reflection orbifolds. We also show that a reflection orbifold covered by the complement of an AP knot is necessarily small. Thus when $K$ is an AP knot, the complement of $K$ covers a reflection orbifold exactly when $K$ is either th…
In previous work, we defined extended versions of von Neumann dimension for Banach space representations of sofic groups. The main application was a definition of l^{p}-dimension and, using this, a proof that the actions of a countable discrete group G on l^{p}(G)^{oplus n} are pairwise non-isomorphic. We answer most of the Conjectures stated at the end of the previous paper "An l^{p}-Version of von Neumann Dimension for Banach Space Representation of Sofic Groups." Tackling the case of finte-dimensional representations involved passing to equivalence relations. This is part of current researc…
The Product Conjecture for the homological Bieri-Neumann-Strebel-Renz invariants is proved over a field. Under certain hypotheses the Product Conjecture is shown to also hold over Z, even though D. Schuetz has recently shown that the Conjecture is false in general over Z. Our version over Z is applied in a joint paper with D. Kochloukova to derive new information about subgroups of Thompson's group F, namely that F has subgroups F_m which are not of type F_{m+1}.
