In this section, we will give some results which show that not just balls but geodesics in an X -graph behave in a homogeneous way when they are relatively distant from the base point, assuming IB(52δ).
Proposition 4.7.1. SupposeΓ is an X -graph with base point ˆb which is∆-vertex- hyperbolic. Suppose thatΓhas IB(∆+1) with constant K, that w labels a geodesic
that lies entirely outside of BK(ˆb), and thatγ is any other path labelled by w and lying entirely outside of BK(ˆb). Thenγis a geodesic.
Proof. Suppose the conclusion is false, and suppose the geodesic that w labels starts
from ˆp andγstarts from ˆq. Let w=w1aw2, where w1is the longest subword which
does label a geodesic starting at ˆq, and a is a word of length 1. Let w′1be a the label of a geodesic such that ˆq·w′1=qˆ·w1a, so that we must have|w′1| ≤ |w1|.
Then we have a geodesic triangle with corners ˆq, ˆq·w1and ˆq·w′1and the obvious
sides connecting them. Let n :=|w1|, and for 0 ≤i <n, let ˆpi :=qˆ·w(i) and
ˆ
qi:=qˆ·w′1(i). Let ˆpn:=qˆ·w1and ˆqn:=qˆ·w′1. This is illustrated in Figure 4.12.
Now, since the triangle above is ∆-thin, we can pick, for each i, a word hi la-
belling a path from ˆpi and ˆqi with|hi| ≤∆. Now we find that for 0≤i<n, each
quadrilateral with corners ˆpi, piˆ+1, ˆqi, qiˆ+1 lies within∆+1 of ˆpi, hence it is con-
tained inside the∆+1-ball around ˆq·w(i), which is isomorphic to the ∆+1-ball around ˆp·w(i)(since this vertex is at a distance of at least K from H).
Using a simple induction, we have ˆp·w1a=pˆ·w′1. But this is a clear contra-
diction, since|w1a|>|w′1|, and w1a labels a geodesic path starting at ˆp. Hence no
such w′1existed, and w labels a geodesic starting at ˆq.
By substituting the point 1 in the group Cayley graph for ˆp in the above argu-
ment, we derive the following similar result:
Proposition 4.7.2. Suppose that G is a finitely generated group, that H is a sub- group with coset Cayley graphΓ′which is∆-vertex-hyperbolic and has GIB(∆+1)
with constant K. If w is a shortest word representing some group element then any path inΓ′labelled by w which lies outside of BK(H)is a geodesic.
It’s a well-known result that in hyperbolic spaces, quasigeodesic paths lie close to geodesic paths, so that if geodesic-labelling words in the Cayley graph label geodesics in the coset Cayley graph when they lie outside a certain radius, the same must be true of quasigeodesics (although the radius in question might be larger).
We see the emergence of one “bad” ball, centred at H in the coset Cayley graph.
4.8
Conclusion and Possible Further Work
This chapter has demonstrated that in the setting of X -graphs, an X -map with qua- siconvex f−1(f(aˆ))preserves a variety of properties.
In Section 4.3 it was pointed out that at least some of these facts are not true for general graphs, but it may be that they generalise to more specific classes, like regular graphs (ie. those graphs in which every vertex has the same valency). One expects that a 2k-regular graph ought to admit edge labels and directions to make it into an X -graph, and that labelling ought to lift through a graph morphism so that hyperbolicity would be preserved in the case of 2k-regular graphs. It would seem more difficult to do this in a way which would preserve (labelled) isomorphisms of balls, however.
Similarly, one might ask whether some of the results can be expanded somehow to general hyperbolic spaces. If the spaces embed X -graphs in a nice way, this would indeed seem to be the case. What about more general spaces?
Chapter 5
Hyperbolic Groups are 14-hyperbolic
The constant of hyperbolicity of a word-hyperbolic group is dependent on its gen- erating set. For example, a free group, say F =<a>on a free generating set has a vertex hyperbolicity constant of 0. However introducing a redundant generator will increase this constant, for example F =<a,b|a2=b> has vertex hyperbolicity constant of 1. The purpose of this chapter is to investigate the lower bound minδ(G) of this constant for a given group G.It turns out that there is a single small such bound that applies to all word- hyperbolic groups. Thus, the value of minδ(G)partitions word-hyperbolic groups into a small number of classes. The bounds given here are likely not to be the smallest due to the naive way in which they are derived, however it is the existence of such a bound that is interesting.
5.1
Thinness of Quasigeodesic Triangles
We first show that if we are working in a geodesic metric space in which all geodesic triangles are δ-vertex-thin and we are given a triangle whose sides are all (1,k)-
quasigeodesics, then the triangle is∆-vertex-thin for some ∆depending only on k andδ.
It is well known that in hyperbolic spaces, quasigeodesic paths lie close to geodesic paths; let us briefly investigate the case of (1,k)-quasigeodesics in par- ticular.
γ α ˆ x ˆ q ˆ p ˆ y ˆ q′
Figure 5.1:(1,k)-quasigeodesics lie close to geodesics
Lemma 5.1.1. Suppose that Γ is a δ-vertex-hyperbolic graph, that γ is a (1,k)- quasigeodesic inΓ joining the vertices ˆx and ˆy, and thatαis a geodesic joining ˆx and ˆy.
Then for every vertex ˆp onγ, there exists a vertex ˆq on α such that d(pˆ,qˆ)≤
k+1
2 +δand d(xˆ,qˆ)≤dγ(xˆ,pˆ)≤d(xˆ,qˆ) + 3k+1
2 .
Proof. Pick geodesics [pˆ,xˆ] and [pˆ,yˆ], and define a geodesic triangle using these andα, as in Figure 5.1. Let m be the meeting point on [pˆ,xˆ]. Then m must be of distance at most2k from ˆp, since
d(pˆ,m) = d(pˆ,xˆ) +d(pˆ,yˆ)−d(xˆ,yˆ) 2 ≤ dγ(pˆ,xˆ) +dγ(2pˆ,yˆ)−d(xˆ,yˆ) = dγ(xˆ,yˆ)−d(xˆ,yˆ) 2 ≤ k 2.
If m lies on a vertex, let ˆq′=m, and if not, let e be the edge containing m and let
ˆ
q′be the vertex on e that is closest to ˆx. Either way, d(qˆ′,m)≤ 12.
Let ˆq be the vertex onαwhich corresponds to ˆq′. Then
d(pˆ,qˆ) ≤ d(pˆ,m) +d(m,qˆ′) +d(qˆ′,qˆ)
≤ k2+1 2+δ,
αxz axz bxz axy αxy αyz ˆ p′ ˆ q′ ˆ q ˆ x ˆ p ˆ y ˆz bxy ˆ p′′
Figure 5.2:(1,k)-quasigeodesic triangles are thin
and d(xˆ,qˆ) ≤ d(xˆ,pˆ) ≤ dγ(xˆ,pˆ) ≤ d(xˆ,pˆ) +k ≤ d(xˆ,qˆ′) +d(qˆ′,pˆ) +k ≤ d(xˆ,qˆ) +3k+1 2 as required.
When the pathsγandαare understood, we will refer to ˆq in Lemma 5.1.1 as the partner of ˆp.
Lemma 5.1.2. SupposeΓis aδ-vertex-hyperbolic graph. Let k be a positive integer, let ˆx, ˆy and ˆz be vertices inΓand letαxy,αyzandαxzbe(1,k)-quasigeodesics joining
ˆ
x to ˆy, ˆy to ˆz and ˆx to ˆz respectively to form a triangleα. Thenαis 3k+3δ+2-vertex-thin.
triangleβ. Let axy, ayzand axz be the meeting points onαand let bxz, byzand bxzbe
the meeting points onβ. See Figure 5.2.
Let ˆp∈αxy be a vertex corresponding to ˆq∈αxz, so dαxz(xˆ,qˆ) =dαxy(xˆ,pˆ). Let
ˆ
p′∈[xˆ,yˆ]and ˆq′∈[xˆ,ˆz]be their respective partners, as in Lemma 5.1.1. By Lemma 5.1.1, the distances d(pˆ,pˆ′)and d(qˆ,qˆ′)are less than or equal toδ+k+21.
By relabelling the corners of the triangle, any pair of corresponding vertices ˆp
and ˆq can be made to fit the above construction. If d(xˆ,pˆ′)>d(xˆ,qˆ′)then swapping ˆ
y and ˆz, and ˆp and ˆq reverses the inquality, so it may be assumed that d(xˆ,pˆ′)≤
d(xˆ,qˆ′).
Suppose d(xˆ,pˆ′)≤d(xˆ,bxy), and let ˆp′′ be the point on [xˆ,ˆz] corresponding to
ˆ
p′, so d(pˆ′,pˆ′′)≤δ. Using the second part of Lemma 5.1.1, we have
d(qˆ′,pˆ′′) = |d(xˆ,qˆ′)−d(xˆ,pˆ′′)|
= |d(xˆ,qˆ′)−d(xˆ,pˆ′)| ≤ 3k2+1,
as dαxz(xˆ,qˆ) =dαxy(xˆ,pˆ). Application of the triangle inequality gives
d(pˆ,qˆ) ≤ d(pˆ,pˆ′) +d(pˆ′,pˆ′′) +d(pˆ′′,qˆ′) +d(qˆ′,qˆ) ≤ k+1 2 +δ +δ+3k+1 2 + k+1 2 +δ = 5k+3 2 +3δ≤3k+3δ+2, as required.
than d(xˆ,bxy). Note that dαxy(xˆ,pˆ) =dαxz(xˆ,qˆ)≤dαxy(xˆ,axz). Then d(xˆ,bxy) < d(xˆ,pˆ′) ≤ dαxy(xˆ,pˆ) ≤ dαxy(xˆ,axy) = dαxy(xˆ,yˆ) +dαxz(xˆ,ˆz)−dαyz(yˆ,ˆz) 2 ≤ d(xˆ,yˆ) +d(xˆ,ˆz2) +2k−d(yˆ,ˆz) = d(xˆ,bxy) +k, so d(pˆ′,bxy) =d(xˆ,pˆ′)−d(xˆ,bxy)≤k and d(pˆ,bxy)≤d(pˆ,pˆ′) +d(pˆ′,bxy)≤k+21+
δ+k= 3k2+1+δ. By symmetry, d(qˆ,bxz)≤ 3k2+1+δalso, so we have d(pˆ,qˆ) ≤ d(pˆ,bxy) +d(bxy,bxz) +d(bxz,qˆ) ≤ 3k+1 2 +δ + (δ+1) + 3k+1 2 +δ = 3k+3δ+2.