Related papers: Realizable ranks of joins and intersections of sub…

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…

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…

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…

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…

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…

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…

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…

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 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…

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…

We answer a question due to A. Myasnikov by proving that all expected ranks occur as the ranks of intersections of finitely generated subgroups of free groups.

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,…

We show that the number of conjugacy classes of intersections $A\cap B^g$, for fixed finitely generated subgroups $A, B<F$ of a free group, is bounded above in terms of the ranks of $A$ and $B$; this confirms an intuition of Walter Neumann.…

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…

Let $G$ be a countable group that is the fundamental group of a graph of groups with finite edge groups and vertex groups that satisfy the strong Atiyah conjecture over $K \subseteq \mathbb{C}$ a field closed under complex conjugation.…

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 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})…

We obtain a number of finiteness results for groups acting on Gromov-hyperbolic spaces. In particular we show that a torsion-free locally quasiconvex hyperbolic group has only finitely many conjugacy classes of $n$-generated one-ended…

We investigate topological realizations of higher-rank graphs. We show that the fundamental group of a higher-rank graph coincides with the fundamental group of its topological realization. We also show that topological realization of…

Let $\mathrm{ex}(n,H,\mathcal{F})$ be the maximum number of copies of $H$ in an $n$-vertex graph which contains no copy of a graph from $\mathcal{F}$. Thinking of $H$ and $\mathcal{F}$ as fixed, we study the asymptotics of…