An infinite left word in some alphabet

Question

What is the infinite left word in some alphabet? I can understand the definition of the infinite right word -- some path in Cayley graph from $id$-element to $+\infty$ or $-\infty$. But is the infinite left word like a path? I think, that it is inverse word for infinite right word, but how I can create a path in Cayley graph, that starts from the infinity and going in the $id$-element?

Answer 1

An infinite word on the alphabet $A$ is an infinite sequence of elements of $A$, usually denoted $$ a_0a_1 \cdots a_n \dotsm $$ A left infinite word is the form $$ \dotsm a_{-n}\dotsm a_{-1}a_0 $$ If you work on the free group of base $A$, the alphabet is $A \cup \bar A$, where $\bar A = \{a^{-1} \mid a \in A\}$.

You can associate to each (right) infinite word $u$ on the alphabet $A \cup \bar A$ the unique infinite path of the Cayley graph starting in $1$ and labelled by $u$. Note however that if $u$ is not a reduced word you may visit several times the same node of the Cayley graph.

You can also associate each left infinite word $\dotsm a_{-n}\dotsm a_{-1}a_0$ the unique path starting in $1$ and labelled by $a_0^{-1}a_1^{-1} \cdots a_n^{-1} \dotsm$.

Finally, note that the term "infinity" requires some care. See End of groups for more details on this question.