Related papers: The Hanna Neumann Conjecture is true when one subg…

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…

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.

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…

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…

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…

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…

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…

We develop a theory of polymatroids on Stallings core graphs, which provides a new technique for proving lower bounds on stable invariants of words and subgroups in free groups $F$, and for upper bounds on their probability for mapping,…

Let $F_n$ be a free group of finite rank $n \geq 2$. We prove that if $H$ is a subgroup of $F_n$ with $\textrm{rk}(H)=2$ and $R$ is a retract of $F_n$, then $H \cap R$ is a retract of $H$. However, for every $m \geq 3$ and every $1 \leq k…

The Friedman--Mineyev theorem, earlier known as the (strengthened) Hanna Neumann conjecture, gives a sharp estimate for the rank of the intersection of two subgroups in a free group. We obtain an analogue of this inequality for any two…

Let $H, K$ be two finitely generated subgroups of a free group, let $\langle H, K \rangle$ denote the subgroup generated by $H, K$, called the join of $H, K$, and let neither of $H$, $K$ have finite index in $\langle H, K \rangle$. We prove…

We explore transversals of finite index subgroups of finitely generated groups. We show that when $H$ is a subgroup of a rank $n$ group $G$ and $H$ has index at least $n$ in $G$ then we can construct a left transversal for $H$ which…

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…

In [3] is was shown that for any group $G$ whose rank (i.e., minimal number of generators) is at most 3, and any finite index subgroup $H\leq G$ with index $[G:H]\geq rank(G)$, one can always find a left-right transversal of $H$ which…

We give new and improved results on the freeness of subgroups of free profinite groups: A subgroup containing the normal closure of a finite word in the elements of a basis is free; Every infinite index subgroup of a finitely generated…

We study the intersection of finitely generated subgroups of free groups by utilizing the method of linear programming. We prove that if $H_1$ is a finitely generated subgroup of a free group $F$, then the WN-coefficient $\sigma(H_1)$ of…

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…

The equaliser of a set of homomorphisms $S: F(a, b)\rightarrow F(\Delta)$ has rank at most two if $S$ contains an injective map, and is not finitely generated otherwise. This proves a strong form of Stallings' Equaliser Conjecture for the…

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})…

A graph $H$ is said to be positive if the homomorphism density $t_H(G)$ is non-negative for all weighted graphs $G$. The positive graph conjecture proposes a characterisation of such graphs, saying that a graph is positive if and only if it…