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\section*{Lecture 12}

\subsection*{Small Cancellation Theory}
\begin{definition}
Let $D\to X$ be a disk diagram in some $2$-complex. A cancelable
pair of $2$-cells in $D$ is a pair of $2$-cells $R_1$, $R_2$ meeting
along a $1$-cell $e$ in $D$ such that $\partial R_1$, $\partial R_2$
starting at $e$ project to the same path in $X$.
\end{definition}
For instance, if $R_1$, $R_2$ fold together along the way to $X$, the
$R_1$, $R_2$ are the cancelable pair.

(figure: a near-immersion with a cancelable pair)

When a presentation has duplicate relators (e.g.
$\langle a,b|abb,abb,babaa\rangle$) it is natural to have
near-immersions with cancelable pairs.
With $X=\langle a,b|(ab)^3\rangle$, $\widetilde{X}$ has many
duplicate $2$-cells.

(figure: tree of hexagons)

{\sc Exercise. }Show $X$ has an index $6$ subgroup that is free
(degree $6$ covering map).

\begin{definition}
A disc diagram $D\to X$ is reduced if it has no cancelable pairs.
\end{definition}

{\sc Note. }If $X$ has no duplicate $2$-cells, no $2$-cells with
periodic attaching maps, then disc diagrams in $X$ are reduced iff
$D\to X$ is a near-immersion.

{\sc Exercise. }If a disc diagram satisfies negative weight test,
then it has no cancelable pairs.

{\sc Note. }Any minimal area disc diagram with boundary path $p$
is reduced, since we can remove cancelable pairs to decrease area.
Converse does not hold. Relationship in $2$ dimensions is not
understood, counterexamples are not aspherical. (figure: cubes)

\subsubsection*{Small Cancelation Conditions}
The maximal internal arcs in reduced disc diagrams are called pieces.

\begin{definition}
$X$ satisfies $C(p)$ condition if for any reduced disc diagram $D\to X$
no boundary path of a $2$-cell $R$ in $D$ is concatenation of $<p$ pieces.
\end{definition}
In other words, reduced daisy with less than $p$ petals doesn't exist.

(figure: daisy flower)

If $X$ is $C(p)$ then in any reduced disc diagram $D\to X$ when we ignore
valence $2$ vertices in $D$, each $2$-cell of $D$ without boundary
(lying in $\partial D$) $1$-cells looks like $\ge p$-gon.

(figure: rectangular figure) $X$ might be $C(4)$, not $C(5)$.

\begin{definition}
$X$ satisfies $T(q)$ condition if each reduced disc diagram
$D\to X$ has the property thattach internal $0$-cell $v\in D$
satisfies either $\deg v=2$ or $\deg v\ge q$.
\end{definition}
(figure: pizza pie) Equivalently,
any reduced pizza pie $\to X$ has $2$ slices, or $\ge q$ slices.

These definitions usually presume that attaching maps of $2$-cells are
immersions.

{\sc Example. }$\langle a,b|abb^{-1}\rangle$

(figure: one-arrowed clock) $T(2)$ --- attaching maps are immersions.

{\sc Note. }$T(q)$ implies immersed attaching maps for $q\ge2$.

We refer to a presentation as $C(p)-T(q)$ if its standard $2$-complex
satisfies $C(p)-T(q)$.

{\sc Note. }A piece in a presentation is a word $p$ that appears in
``more than one way'' among relators.

{\sc Example. }$\langle a,b|abbaaabbaaab\rangle$ --- $aa$, $abbaaab$.

$\langle a|aaaaaaa\rangle$ is $C(\infty)-T(\infty)$. For instance,
a diagram corresponding to piece $aaa$ is not reduced. (figure: a diagram)

$\langle a,b,c| (abc)^{100}\rangle$ is $C(\infty)-T(\infty)$.

{\sc Exercise. }Every group has a $C(5)$ presentation (also has $C(4)-T(3)$
and $C(3)-T(4)$ presentations). It's about subdivision.

\begin{definition}
$X$ satisfies $C'(\alpha)$ condition if for each piece $p$ occurring
in $\partial_pR$ we have $|p|<\alpha|\partial_pR|$. (This is a metric
small-cancelation condition.)
\end{definition}

{\sc Note. }$C'(\frac{1}{n})\Rightarrow C(n+1)$.

{\sc Example. }
$\langle a,b|abab^2ab^3\cdots ab^{100}, baba^2ba^3\cdots ba^{100}\rangle$.
A piece is not longer than $b^{98}ab^{99}$, so it's
$C'(\sim \frac{4}{99} )\Rightarrow C(6)$.

\end{document}
