10 research outputs found
Limit trees for free group automorphisms: universality
To any free group automorphism, we associate a universal (cone of) limit
tree(s) with three defining properties: first, the tree has a minimal isometric
action of the free group with trivial arc stabilizers; second, there is a
unique expanding dilation of the tree that is equivariant with respect to the
automorphism; and finally, the loxodromic elements are exactly the elements
that weakly limit to topmost attracting laminations under forward iteration by
the automorphism. So the action on the tree detects the automorphism's topmost
exponential dynamics.
As a corollary, our previously constructed limit pretree that detect the
exponential dynamics is canonical; the pretree admits pseudometrics that can be
viewed as a universal hierarchy of limit trees. For atoroidal automorphisms,
this universal hierarchical decomposition is analogous to the Nielsen--Thurston
normal form for a surface homeomorphism or the Jordan canonical form for a
linear map. We use it to sketch a proof that atoroidal outer automorphisms have
virtually abelian centralizers.Comment: 58 pages, 3 figure
Hyperbolic Endomorphisms of Free Groups
We prove that ascending HNN extensions of free groups are word-hyperbolic if and only if they have no Baumslag-Solitar subgroups. This extends Brinkmann\u27s theorem that free-by-cyclic groups are word-hyperbolic if and only if they have no Z2 subgroups. To get started on our main theorem, we first prove a structure theorem for injective but nonsurjective endomorphisms of free groups. With the decomposition of the free group given by this structure theorem, we (more or less) construct representatives for nonsurjective endomorphisms that are expanding immersions relative to a homotopy equivalence. This structure theorem initializes the development of (relative) train track theory for nonsurjective endomorphisms
The minimal genus problem for right angled Artin groups
We investigate the minimal genus problem for the second homology of a right
angled Artin group (RAAG). Firstly, we present a lower bound for the minimal
genus of a second homology class, equal to half the rank of the corresponding
cap product matrix. We show that for complete graphs, trees, and complete
bipartite graphs, this bound is an equality, and furthermore in these cases the
minimal genus can always be realised by a disjoint union of tori. Additionally,
we give a full characterisation of classes that are representable by a single
torus. However, it is not true in general that the minimal genus of a second
homology class of a RAAG is necessarily realised by a disjoint union of tori:
we construct a genus two representative for a class in the pentagon RAAG.Comment: 19 pages, 4 figures; comments welcom
Hyperbolic hyperbolic-by-cyclic groups are cubulable
We show that the mapping torus of a hyperbolic group by a hyperbolic
automorphism is cubulable. Along the way, we (i) give an alternate proof of
Hagen and Wise's theorem that hyperbolic free-by-cyclic groups are cubulable,
and (ii) extend to the case with torsion Brinkmann's thesis that a torsion-free
hyperbolic-by-cyclic group is hyperbolic if and only if it does not contain
-subgroups.Comment: 11 page
Limit pretrees for free group automorphisms: existence
To any free group automorphism, we associate a real pretree with several nice
properties. First, it has a rigid/non-nesting action of the free group with
trivial arc stabilizers. Secondly, there is an expanding pretree-automorphism
of the real pretree that represents the free group automorphism. Finally and
crucially, the loxodromic elements are exactly those whose (conjugacy class)
length grows exponentially under iteration of the automorphism; thus, the
action on the real pretree is able to detect the growth type of an element.
This construction extends the theory of metric trees that has been used to
study free group automorphisms. The new idea is that one can equivariantly blow
up an isometric action on a real tree with respect to other real trees and get
a rigid action on a treelike structure known as a real pretree. Topology plays
no role in this construction as all the work is done in the language of
pretrees.Comment: 41 pages, 1 figure. v2: 40 pages, referenced the seque
Hyperbolic hyperbolic-by-cyclic groups are cubulable
11 pagesWe show that the mapping torus of a hyperbolic group by a hyperbolic automorphism is cubulable. Along the way, we (i) give an alternate proof of Hagen and Wise's theorem that hyperbolic free-by-cyclic groups are cubulable, and (ii) extend to the case with torsion Brinkmann's thesis that a torsion-free hyperbolic-by-cyclic group is hyperbolic if and only if it does not contain -subgroups