Ascending HNN Extensions Research Articles

Orbit problems for free-abelian times free groups and related families

We prove that the Brinkmann Problems ( BrP & BrCP ) and the Twisted-Conjugacy Problem ( TCP ) are decidable for any endomorphism of a free-abelian times free (FATF) group [Formula: see text]. Furthermore, we prove the decidability of the two-sided Brinkmann Conjugacy Problem ( 2BrCP ) for monomorphisms of FATF groups (and combine it with TCP ) to derive the decidability of the conjugacy problem for ascending HNN extensions of FATF groups.

Lifting Semistability in Finitely Generated Ascending HNN-Extensions

Lifting Semistability in Finitely Generated Ascending HNN-Extensions

Elementary amenable groups of cohomological dimension 3

Abstract We show that torsion-free elementary amenable groups of Hirsch length at most 3 are solvable, of derived length at most 3. This class includes all solvable groups of cohomological dimension 3. We show also that groups in the latter subclass are either polycyclic, semidirect products BS ⁢ ( 1 , n ) ⋊ Z \mathrm{BS}(1,n)\rtimes\mathbb{Z} , or properly ascending HNN extensions with base Z 2 \mathbb{Z}^{2} or π 1 ⁢ ( Kb ) \pi_{1}(\mathrm{Kb}) .

Failure of the finitely generated intersection property for ascending HNN extensions of free groups

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 groups 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.

The dynamics and geometry of free group endomorphisms

We prove that ascending HNN extensions of free groups are word-hyperbolic if and only if they have no Baumslag-Solitar subgroups. This extends the theorem of Brinkmann that free-by-cyclic groups are word-hyperbolic if and only if they have no free abelian subgroups of rank 2. The paper is split into two independent parts:1) We study the dynamics of injective nonsurjective endomorphisms of free groups. We prove a canonical structure theorem that initializes the development of improved relative train tracks for endomorphisms; this structure theorem is of independent interest since it makes many open questions about injective endomorphisms tractable.2) As an application of the structure theorem, we are able to (relatively) combine Brinkmann's theorem with our previous work and obtain the main result stated above. In the final section, we further extend the result to HNN extensions of free groups over free factors.

Irreducibility of a free group endomorphism is a mapping torus invariant

We prove that the property of a free group endomorphism being irreducible is a group invariant of the ascending HNN extension it defines. This answers a question posed by Dowdall–Kapovich–Leininger. We further prove that being irreducible and atoroidal is a commensurability invariant. The invariance follows from an algebraic characterization of ascending HNN extensions that determines exactly when their defining endomorphisms are irreducible and atoroidal; specifically, we show that the endomorphism is irreducible and atoroidal if and only if the ascending HNN extension has no infinite index subgroups that are ascending HNN extensions.

Bounded Depth Ascending HNN Extensions and -Semistability at infinity

AbstractA well-known conjecture is that all finitely presented groups have semistable fundamental groups at infinity. A class of groups whose members have not been shown to be semistable at infinity is the class ${\mathcal{A}}$ of finitely presented groups that are ascending HNN-extensions with finitely generated base. The class ${\mathcal{A}}$ naturally partitions into two non-empty subclasses, those that have “bounded” and “unbounded” depth. Using new methods introduced in a companion paper we show those of bounded depth have semistable fundamental group at infinity. Ascending HNN extensions produced by Ol’shanskii–Sapir and Grigorchuk (for other reasons), and once considered potential non-semistable examples are shown to have bounded depth. Finally, we devise a technique for producing explicit examples with unbounded depth. These examples are perhaps the best candidates to date in the search for a group with non-semistable fundamental group at infinity.

HNN decompositions of the Lodha–Moore groups, and topological applications

The Lodha–Moore groups provide the first known examples of type [Formula: see text] groups that are non-amenable and contain no non-abelian free subgroups. These groups are related to Thompson’s group [Formula: see text] in certain ways, for instance they contain it as a subgroup in a natural way. We exhibit decompositions of four Lodha–Moore groups, [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], into ascending HNN extensions of isomorphic copies of each other, both in ways reminiscent to such decompositions for [Formula: see text] and also in quite different ways. This allows us to prove two new topological results about the Lodha–Moore groups. First, we prove that they all have trivial homotopy groups at infinity; in particular they are the first examples of groups satisfying all four parts of Geoghegan’s 1979 conjecture about [Formula: see text]. Second, we compute the Bieri–Neumann–Strebel invariant [Formula: see text] for the Lodha–Moore groups, and get some partial results for the Bieri–Neumann–Strebel–Renz invariants [Formula: see text], including a full computation of [Formula: see text].

Strictly ascending HNN extensions in soluble groups

We show that there exist finitely generated soluble groups which are not LERF but which do not contain strictly ascending HNN extensions of a cyclic group. This solves Problem 16.2 in the Kourovka notebook. We further show that there is a finitely presented soluble group which is not LERF but which does not contain a strictly ascending HNN extension of a polycyclic group.

GROUP RINGS OF COUNTABLE NON-ABELIAN LOCALLY FREE GROUPS ARE PRIMITIVE

We prove that every group ring of a non-abelian locally free group which is the union of an ascending sequence of free groups is primitive. In particular, every group ring of a countable non-abelian locally free group is primitive. In addition, by making use of the result, we give a necessary and sufficient condition for group rings of ascending HNN extensions of free groups to be primitive, which extends the main result in [Group rings of proper ascending HNN extensions of countably infinite free groups are primitive, J. Algebra317 (2007) 581–592] to the general cardinality case.

Filtered Ends, Proper Holomorphic Mappings of Kähler Manifolds to Riemann Surfaces, and Kähler Groups

The main result of this paper is that a connected bounded geometry complete Kahler manifold which has at least 3 filtered ends admits a proper holomorphic mapping onto a Riemann surface. As an application, it is also proved that any properly ascending HNN extension with finitely generated base group, as well as Thompson’s groups V, T, and F, are not Kahler. The results and techniques also yield a different proof of the theorem of Gromov and Schoen that, for a connected compact Kahler manifold whose fundamental group admits a proper amalgamated product decomposition, some finite unramified cover admits a surjective holomorphic mapping onto a curve of genus at least 2.

Group rings of proper ascending HNN extensions of countably infinite free groups are primitive

We prove that every group ring of a proper ascending HNN extension of a free group with at most countably infinite rank is primitive.

Ascending HNN extensions of polycyclic groups are residually finite

We prove that every ascending HNN extension of a polycyclic-by-finite group is residually finite. We also give a criterion for the residual finiteness of an ascending HNN extension of a residually nilpotent group, and apply this criterion to recover a result of Moldavanskiı̆ on the residual finiteness of certain ascending HNN extensions of free groups.

Ascending HNN extensions of residually finite groups can be non-Hopfian and can have very few finite quotients

We produce two examples demonstrating that an ascending HNN extension of a finitely generated residually finite group need not be residually finite. The first example has very few quotients. The second example is not Hopfian.

ASCENDING HNN EXTENSIONS OF FINITELY GENERATED FREE GROUPS ARE HOPFIAN

It is shown that every ascending HNN extension of a finitely generated free group is Hopfian. An important ingredient in the proof is that under certain hypotheses on the group H, if G is an ascending HNN extension of H, then cd(G) = cd(H) + 1.