All Questions
Tagged with cayley-graphs geometric-group-theory
11
questions
20
votes
4
answers
2k
views
Cayley graph of $A_5$ with generators $(1,2,3,4,5),(1,4,3,2,5)$
The Cayley graph of $A_5$ with two generators of order 5 seems rather complicated. What is its graph genus (orientable or non-orientable)?
The best I could get by trial and error is an embedding ...
- 24.5k
16
votes
0
answers
311
views
Does every infinite, connected, locally finite, vertex-transitive graph have a leafless spanning tree?
My question is
Let $G$ be an infinite, connected, locally finite, vertex-transitive graph. Must
$G$ have the following substructures?
i) a leafless spanning
tree;
ii) a spanning forest consisting ...
- 1,844
15
votes
0
answers
238
views
Approximation of the effective resistance on Cayley graph
Let $\Gamma$ be a finitely generated group, and denote by $G$ the Cayley graph of $\Gamma$. Denote by $d_R$ the resistance distance metric on this graph. The resistance distance metric between the ...
- 385
8
votes
2
answers
2k
views
Quasi-isometries vs Cayley Graphs
The following questions might be trivial, however, I couldn't solve them:
Let $G$ be generated by a finite symmetric set $S$. Suppose that $\Gamma(G,S)$ is the corresponding right Cayley graph of $G$...
- 244
6
votes
1
answer
137
views
Does the visual boundary of any one-ended Cayley graph contain at least three points?
Let $\Gamma$ be a Cayley graph of a finitely generated group. We can define the visual boundary of $\Gamma$ with respect to some base vertex $b$, denoted $\partial \Gamma$, as the set of geodesic rays ...
- 537
6
votes
1
answer
244
views
Is the function $k(g,h) = \frac{1}{1+\lvert gh^{-1}\rvert}$ positive definite?
Let $G$ be a finite group, $S \subset G$ a generating set, closed under taking inverses, and $\lvert\cdot\rvert$ the word length with respect to this set $S$.
Question. Is the function $k(g,h) = \...
- 2,126
5
votes
2
answers
751
views
A generously vertex transitive graph which is not Cayley?
A graph is vertex transitive if $x \mapsto y$ by an automorphism.
A graph is generously vertex transitive if $x \mapsto y \mapsto x$ by an automorphism.
Simple facts:
GVT $\rightarrow$ unimodular. ...
- 153
5
votes
1
answer
374
views
Cayley graph properties
Consider an infinite graph that satisfies the following property: if any finite set of vertices is removed (and all the adjacent edges), then the resulting graph has only one infinite connected ...
3
votes
0
answers
357
views
What about a Cayley n-complex for n>2?
Let $G$ be a finitely presented group. The Cayley graph of the finite generating set is a $1$-complex where the $0$-cells are the elements of $G$ and the $1$-cells are given by the generators (...
- 25.5k
3
votes
0
answers
289
views
Induced graphs of Cayley graph
I have a Cayley graph $\mathrm{Cay}(G,S)$, its group presentation $G=\langle S | R \rangle$, and it becomes a metric graph by assigning a length equal to $1$ to each edge. I also have an induced ...
- 31
0
votes
0
answers
74
views
Distances on spheres in Cayley graphs of non-amenable groups
Let $G$ be a non-amenable group (or perhaps more generally, a group with exponential growth). For any $\epsilon>0$, define the shell of radius r, $S_\epsilon(r)$, as the set of points that lie at a ...
- 21