On Cayley graphs: Part 1
A Cayley graph can be considered as an attempt to geometrize or visualize groups. Algebraically, a group is just a bunch of letters some of them having an inverse script. By visualizing groups, we may observe some patterns in their structure. So let's see how can we construct Cayley graphs. Take an arbitrary group $G$ with a set of generators $S\subset G$. In other words, every element of $G$ can be written as:
\begin{equation}
g = s_1^{\epsilon_1}s_2^{\epsilon_2}\ldots s_n^{\epsilon_n},\quad \epsilon_i\in\{\pm1 \}
\end{equation}
As for $S$, we assume that $S^{-1} = S$, i.e. $s\in S$ implies $s^{-1}\in S$, and $1\not\in S$. In fact these assumptions are not necessary, but tend to make life a bit easier. Let us construct a Cayley graph $\Gamma=(V,E)$ as follows:
We can observe few things straight away about $\Gamma$. It is undirected as $gs = h$ happens when $g = hs^{-1}$. Any two vertices are connected. Now, if $S$ is an infinite set, then each node should have an infinite number of edges. For people from mathematical logic who work with infinite trees, where each node has infinite number of offsprings, that should not be a big deal. However generally, we are not accustomed to deal with infinite number of edges for a single node. So we require $S$ to be a finite set, which makes $G$ finitely generated group. Is this a considerable limitation? Perhaps, I do not know to be honest. On the brighter side, there should still be many groups which are finitely generated. On this note, let us end this post. In the next, we plan to transform the graph $\Gamma$ into a metric space $X$ which enjoys action of $G$.
References:
- Alessandro Sisto, Lecture notes on Geometric Group Theory.
\begin{equation}
g = s_1^{\epsilon_1}s_2^{\epsilon_2}\ldots s_n^{\epsilon_n},\quad \epsilon_i\in\{\pm1 \}
\end{equation}
As for $S$, we assume that $S^{-1} = S$, i.e. $s\in S$ implies $s^{-1}\in S$, and $1\not\in S$. In fact these assumptions are not necessary, but tend to make life a bit easier. Let us construct a Cayley graph $\Gamma=(V,E)$ as follows:
- $V=G$
- $(g,h)\in E$ if $gs = h$ for some $s\in S$, or equivalently $g^{-1}h\in S$
We can observe few things straight away about $\Gamma$. It is undirected as $gs = h$ happens when $g = hs^{-1}$. Any two vertices are connected. Now, if $S$ is an infinite set, then each node should have an infinite number of edges. For people from mathematical logic who work with infinite trees, where each node has infinite number of offsprings, that should not be a big deal. However generally, we are not accustomed to deal with infinite number of edges for a single node. So we require $S$ to be a finite set, which makes $G$ finitely generated group. Is this a considerable limitation? Perhaps, I do not know to be honest. On the brighter side, there should still be many groups which are finitely generated. On this note, let us end this post. In the next, we plan to transform the graph $\Gamma$ into a metric space $X$ which enjoys action of $G$.
References:
- Alessandro Sisto, Lecture notes on Geometric Group Theory.
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