All Questions
18 questions
2
votes
1
answer
292
views
Do balls in expander graphs have small expansion?
Consider a $d$-regular infinite transitive expander graph $G$, and let $B_r$ be a ball of radius $r$ in $G$. Can one place any upper bounds on the expansion of $B_r$?
My intuition is that $B_r$ will ...
- 31
1
vote
1
answer
80
views
Deduce unsolvability of $\operatorname{IP}(G_0)$ from the Adian–Rabin Theorem
$\operatorname{IP}(G_0)$: the special isomorphism problem for $G_0$, i.e., given $G_0$, determine if $G$ is isomorphic to $G_0$. My question is that how can we deduce from the Adian–Rabin theorem that ...
- 51
7
votes
1
answer
283
views
Are two quasi-isometric, isomorphic on large enough balls, transitive graphs isomorphic?
Take two transitive graphs $X,Y$ (potentially directed and edge-labelled, e.g. Cayley graphs).
Assume $X,Y$ are quasi-isometric with constant $K$, i.e. there exists a function $f:VX \to VY$ ($VX,\,VY$ ...
- 887
7
votes
1
answer
319
views
Which groups contain a comb?
The comb is the undirected simple graph with nodes
$\mathbb{N} \times \mathbb{N}$
where $\mathbb{N} \ni 0$ and edges
$$ \{\{(m,n), (m,n+1)\}, \{(m,0), (m+1,0)\} \;|\; m \in \mathbb{N}, n \in \mathbb{N}...
- 6,652
7
votes
1
answer
247
views
Going up of an amalgamated decomposition of a subgroup of finite index
Let $G$ be a finitely presented group and H a subgroup of index $n$ in $G$. Suppose that H has a non-trivial decomposition as amalgamated product, say $H = A \ast_U B$. I am wondering about the ...
5
votes
1
answer
407
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 ...
4
votes
0
answers
215
views
Words Growth in Finite Groups
Let $G$ be a finite group with a set of generators and let $\Gamma$ be its Cayley Graph. Let $b_k$ be the number of elements in the ball of radius $k$. I am interested in what is known about the ...
- 5,450
5
votes
0
answers
169
views
In the literature on infinite graphs, are there results on "periodizable" graphs?
Let $G=(V,E)$ be a connected countably infinite $k$-regular simple graph (no loops or multiple edges). For $A$ a finite subset of $V$, let me denote by $G_A=(A,E_A)$ the induced subgraph with vertex ...
8
votes
2
answers
343
views
Cubic almost-vertex-transitive graphs with given spanning tree
Consider the infinite 3-regular tree. Pick a vertex $C$, the "center".
For any integer $L\ge 1$ consider the closed ball, in the graph distance, of radius $L$ around $C$. Let $T_L$ be the induced ...
3
votes
0
answers
285
views
Cayley Graphs and Cyclically reduced words [closed]
Let $G$ be a finite group and $S$ be a symmetric generating set for $G$. (EDIT: Assume $S$ does not contain involutions!) Cyclically reduced words can be thought of as minimal length representatives ...
- 949
4
votes
2
answers
871
views
Detecting HNN-Extension and free products with amalgamation
This question is partly connected with the following Connection between Stalling's end theorem and Seifert-van Kampen Theorem.
By Stalling's Theorem a group with more than one end splits over a ...
- 721
4
votes
1
answer
323
views
Obtaining a quasi-isometry of the 'boundary'
It is well-known that a quasi-isometry induces a homeomorphism on the space of ends of say a locally finite graph for simplicity. Clearly the converse is not true. In other words the concept of ends ...
- 721
5
votes
2
answers
805
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
2
votes
1
answer
346
views
Limit Group decomposition
I would need a clarification about a statement in the article Limit groups and groups acting freely on $\mathbb{R}^n$-trees by Vincent Guirardel.
First recall that a limit group is a finitely ...
- 41
13
votes
1
answer
887
views
Isometries of some simple Cayley graphs
Consider a Cayley graph of a group $G$ with respect to a symmetric finite generating set $S$.
There are some obvious candidates to isometries of this graph - for example, translation by elements of $G$...
- 689
13
votes
1
answer
393
views
Is the Cayley graph of Thompson's group isolated in the space of vertex-transitive graphs?
Consider Thompson's group (the one commonly referred to as $T$), which is finitely presentable. Consider the Cayley graph, but then forget the coloring and direction on edges. So now we just have an ...
- 3,073
3
votes
0
answers
311
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
4
votes
2
answers
1k
views
Hausdorff Dimension of Cayley Graphs of Groups
I was wondering what has been done concerning the Hausdorff measure of the Cayley graphs of finitely generated countable groups. There are number of issues that would need to be dealt with:
1.) By ...
- 2,441