设H和K是秩为h和k的自由群的子群\geq h。证明了Burns不等式的强形式：rank（H \cap K）-1 \leq 2（h-1）（k-1）-（h-1）（rank（H \vee K）-1）.这个推论也是由L. Louder和D. B. McReynolds，已被M. Culler和P. Shalen获得了关于双曲三维流形的体积的信息。我们还证明了下面的特殊情况下的汉娜诺依曼猜想，这也已经获得了劳德。如果H和K的并的秩至少为（h + k + 1）/2，则H和K的交集的秩不超过（h-1）（k-1）+ 1。
英文摘要：
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-1)(k-1) + 1.
The main result in this paper is the failure of the finitely generated intersection property (FGIP) of ascending HNN extensions of non-cyclic finite rank free groups. This class of group consists of free-by-cyclic groups and properly ascending HNN extensions of free groups. We also give a sufficient condition for the failure of the FGIP in the context of relative hyperbolicity, we apply this to free-by-cyclic groups of exponential growth.
We extend the classical Stallings theory (describing subgroups of free groups as automata) to direct products of free and abelian groups: after introducing enriched automata (i.e., automata with extra abelian labels), we obtain an explicit bijection between subgroups and a certain type of such enriched automata, which - as it happens in the free group - is computable in the finitely generated case. This approach provides a neat geometric description of (even non finitely generated) intersections of finitely generated subgroups within this non-Howson family. In particular, we give a geometric s…
Let $G$ be a non-abelian finite simple group. In addition, let $Δ_G$ be the intersection graph of $G$, whose vertices are the proper nontrivial subgroups of $G$, with distinct subgroups joined by an edge if and only if they intersect nontrivially. We prove that the diameter of $Δ_G$ has a tight upper bound of 5, thereby resolving a question posed by Shen (2010). Furthermore, a diameter of 5 is achieved only by the baby monster group and certain unitary groups of odd prime dimension.
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. This result was previously known only in the case where $A$ or $B$ is cyclic by the $w$-cycles theorem of Helfer and Wise and, independently, Louder and Wilton. In order to prove our main theorem, we introduce new results regarding the structure of fibre products of finite graphs. We work with a generalised definition of graph immersions so that our results…
Let F be the free group functor, left adjoint to the forgetful functor between the category of groups GRP and the category of sets SET. Let f from A to B, and h from A to C be two functions in SET and let Ker(F(f)) and Ker(F(h)) be the kernels of the induced morphisms between free groups. Provided that the kernel pairs Eq(f) and Eq(h) of f and h permute (such as it is the case when the pushout of f and h is a double extension in SET), this short article describes a method to rewrite a general element in the intersection of Ker(F(f)) and Ker(F(g)) as a product of generators in A which is (f,h)-…
