All Questions
7 questions
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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 ...
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6
votes
1
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181
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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 ...
- 63.9k
3
votes
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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 (...
6
votes
1
answer
245
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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) = \...
- 12.9k
5
votes
2
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805
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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. ...
- 63.9k
5
votes
1
answer
407
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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 ...
- 28.2k
3
votes
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311
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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 ...
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