Part Three: Comparing Topological Boundaries up to Homeomorphism

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Harmonic Maps on Graphs and Compactification
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Floyd Boundaries are only one of many boundaries defined as completions of metric spaces and using geodesic rays. Comparing these boundaries and their relationships to harmonic maps, group structure, and properties of metric spaces gives rise to some interesting results and questions.


Table of contents
1 Floyd, Ideal and Gromov Boundaries

1.1 Theorem

2 Harmonic Isometries

2.1 References

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Floyd, Ideal and Gromov Boundaries


\partial X_{F}: Floyd boundary with respect to Floyd Admissible Function F

\partial X_{G}: Gromov boundary \partial X_{I}: Ideal boundary

Before giving the comparison results, I will define δ − hyperbolic and CAT(0) space:

In order to define a CAT(0) space, I will define the notion of a triangle map, f: T - \mathbb{R}^{2}, where T is a geodesic triangle. A triangle map has the property that it preserves the length of the sides of T and the restriction of f to each geodesic side in T is an an isometry. Now, a CAT(0) is a geodesic space for which every triangle map has d(u,v) \leq |f(u) - f(v)|, for all u, v \in T.

To define δ − hyperbolic, I first need to define the Gromov inner product with respect to a vertex p \in X:

\left \langle x , y \right \rangle_{p} = (d(x,p) + d(p,y) - d(x,y))/2, \ x, \ y \in X.

We then say that the metric space X is δ − hyperbolic if

\left \langle x, z \right \rangle_{p} \geq min ( \left \langle x, y \right \rangle_{p}, \left \langle y, z \right \rangle_{p} - \delta

which, unlike the triangle inequality, gives a lower bound the distance between x and z.


From [4, p. 8], the following relationships among the boundaries occur:

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Theorem

Assume that the space (graph) X is δ − Hyperbolic and nonempty

Then

(1) Since there is a K > 0 and a ε0(δ) > 0 for which F(t)exp(\epsilon_{0}t) \geq K, then \partial X_{G} \simeq \partial X_{F}, under the Gromov dε metric. Furthermore, if X is complete and CAT(0) then \partial X_{I} \simeq \partial X_{F}.

(2) If X is complete and CAT(0), then \partial X_{I} \simeq \partial X_{G} under the Gromov dε metric.

(3) If X is proper and ε satisfies \epsilon \delta \leq 1/5, then \partial X_{F} \simeq \partial X_{G} under the Gromov dε metric.

A result about comparing Floyd and Hyperbolic Boundaries comes from [5, p. 8] and states that if the geodesic space is the Cayley graph of a finitely generated word hyperbolic group, then \partial X_{F} (for F(r) = (r + 1) − 2) coincides with the standard Hyperbolic boundary.

Note: Karlsson also compares the Floyd boundary of certain groups to a particular limit set: |L_{s}| \leq |\partial X_{F}| = |\partial \Gamma|

where Γ is the fundamental group a hyperbolic 3-manifold and Ls is the limit set of this manifold.


Poisson Boundary

The Poisson boundary also merits comparison, particularly because of its applications in solving the Dirichlet Problem. To define the Poisson boundary, I first need to defined the notion of a Markov Chain induced by a probability measure μ, the path space \Gamma^{\mathbb{Z}_{+}}, and the time Shift T, respectively, deriving all the information from [8, p. 664]

Given, a probability measure μ on a group, the induced random walk is defined as p(x,y) = μ(x − 1y). Note that p(hx,hy) = p(x,y).


The space of sample paths x = {x_{n}}, n \geq 0 under which the group Γ acts coordinate-wise is called the path space.


The time shift in the path space in the path space \Gamma^{\mathbb {Z}_{+}} is defined as a map T: {x_{n}} \rightarrow {x_{n+1}} such that TPθ = θP = Pθμ, where Pm denotes the measure in the path space corresponding to the initial distribution of an element counted m steps away from the identity (since the group Γ was assumed to be countable) and Pθ = θP is dominated by it.


The Poisson Boundary is defined as the space of ergodic components G of the time shift T in the path space (\Gamma^{\mathbb{Z}_{+}}, P_{m}).


Now, we need to introudce Kaimanovich's conditions (CP), (CS), and (CG) :

(CP) or Projectivity Any two sequences remaining a bounded distance from each other must converge to one and the same boundary point. Equivalently, for a group G, if the sequence g_{n} \in G has g_{n} \rightarrow g \in \partial G then g_{n}x \rightarrow g for all x \in G.

(CS) Separated by strips For any two \gamma, \eta \in \partial \Gamma, S(γ,η) be the strip bounded by the geodesics rays eminating from infinity and converging to γ and η respectively. Then the condition states that for any three points, x_{i} \in \partial X, there are neighborhoods Ui about each such that S(U_{1}, U _{2}) \cap U_{3} = \emptyset.

(CG) d is left-invariant, i.e. d(x,y) = d(gx,gy) for all g, h \in \Gamma. and the gauge is temperate. According to Kaimonich, a gauge is a sequence of sets \mathcal{G}_{k}, (k \geq 1) that approximate the entire group. Such a gauge is temperate if sup_{k} log \ \frac{1}{k} \ \mathcal{G}_{k} < \infty.

Karlsson proves that if Γ contains at least three points, then \overline{\Gamma} satisfies (CP),(CS), and (CG) [5. p. 9].

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Harmonic Isometries


Studying these properties about metric and geodesic spaces raises the question: When is a harmonic map on a subset of \overline{X} an isometry, and what does this say about the geometry of \ X, \partial X or the algebra of the group of which X is Cayley Graph. Also, what do these functions tell us about the solvability of the Dirichlet Problem?

Note: The relationship to the Dirichlet Problem can be seen in part from the Karlsson's proposition [2, p.9] that \partial \Gamma is a geodesic space.

By definition an isometry into \mathbb{R} has the following property: d(x,y) = | f(x) − f(y) | , where d denotes the metric on the graph. As we have seen above, this metric could be the induced path metric, the Floyd metric, or the Gromov metric.

In any case, the condition that this isometry be harmonic is


\sum_{y \sim x} f(y) - f(x) = \sum_{y \sim x} sgn_{f}(y,x) d(y,x) = 0,
where \ sgn_{f}(y,x) = sgn (f(y) - f(x)).

Now, in the case that d is the induced path metric, we have the somewhat conciser requirement that \sum_{y \sim x} sgn_{f}(y,x) = 0.

I will call a map that is simultaneously a harmonic and isometric a harmonic isometry.

Proposition

Let X be a graph containing a vertex of odd degree and let d denote the induced path metric. Then X does not admit any harmonic isometries to \mathbb{R}.

Proof:

Let v \in X be the vertex of odd degree and let \ f denote any isometry from \ X to \mathbb{R}. Then for any y \in N(v), \ |f(y) - f(v)| = 1. Therefore,

\sum_{y \in N(v)} sgn \ d(y,v)

will only be zero if there are an equal number of + 1 s and − 1 s. Since \ |N(v)| is odd, this is not possible.


Theorem

Let k be an even integer. Then every k - regular graph admits a harmonic isometry under the induced path metric.


Proof:

We can easily define an isometry on a k - regular graph simply by sending each vertex to its distance from a point chosen to be the center (all of these values will be in \mathbb{Z}.) Denote this isometry by f.

Proceeding by induction on the distance from the center, we see that we can easily assure that \ f is harmonic is the sphere of radius one around the center \ z by negating half of the values. Since the center was taken as the starting point, this map indeeds remains an isometry. Now, assume the function can be extended to a sphere of radius m \geq 1 about the center such that, when restricted to this sphere, it is an isometry and also harmonic at every vertex whose neighborhood lies within the sphere. The sphere of radius m + 1 can be formed by taking the union of all the neighbors of vertices in Sm whose neighbors lay outside of Sm. We need only to make sure that f remain an isometry at these vertices that that its restriction to Sm + 1 is an isomety. Now, let v \in S_{m} such that N(v) \in S_{m+1} - S_{m}. For a regular graph this is actually just the set of vertices for which d(z,v) = m, where z is the center of the graph. Since the restriction of f to Sm is an isometry, the values at the neighbors N(v) differ from f(v) 1. Since the graph is regular of even degree, there must be as many neighbors within Sm as there are in Sm + 1 − Sm. Therefore, we may choose each value f(w) of the neighbors w \in S_{m+1} - S_{m} so that they correspond to a w* \in S_{m} and satisfy f(w) + f(w * ) − 2f(v) = 0. Since each neighbor in Sm + 1 − Sm has distance m + 1 from the center, and each other neighbor has distance m − 1 from the center (since the graph is regular) it follows that this map remains an isometry at vertices lying at a distance of m + 1 from the center is harmonic for all vertices lying at a distance of m from the center. The result thus follows by induction.


Conjecture about the Dirichlet Problem


Discrete Analytic Functions

Extending the Definition of Hamiltonian to Infinite Graphs


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References

[1] Lovasz, Laszlo Discrete Analytic Functions: An Exposition Microsoft Research

[2] Karlsson, Anders Some remarks concerning harmonic functions on homogeneous graphs

Discrete Mathematics and Theoretical Computer Science AC (2003), 137-144

[3] Woess, Wolfgang Dirichlet problem at infinity for harmonic functions on graphs 31C12, 60J15 (1991).

[4] Buckley, Stephen M. and Kokkendorff, Simon L. Comparing the Floyd and Ideal Boundaries of a Metric Space (14.02.2005)

[5] Karlsson, Anders Boundaries and random walks on finitely generated infinite groups Forschunginstitut fur Mathematik CH8092 (2001)

[6] Elek, Gabor and Tardos, Gabor On Roughly Transitive Amenable Graphs and Harmonic Dirichlet Functions Proceedings of the American Matehematical Society Volume 128, Number 89, Pages 2479-2485 (2000).

[7] Flanigan, Francis J. Complex Variables: Harmonic and Analytic Functions (1972).

[8] Kaimanovich, Vadim A. The Poisson formula for groups with hyperbolic properties Annals of Mathematics, 152 (2000), 659-692.

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