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\title[Generic doubly-twisted conjugacy classes]{Generic
  doubly-twisted conjugacy classes in free groups (preliminary)}
\author[Staecker]{P. Christopher Staecker}
\institute[Messiah College]{Messiah College, Grantham PA}
\date[]{Joint Math Meetings, 2009}

\begin{document}

\frame{\titlepage}
\section{Preliminaries}
\frame
{
  \frametitle{Setting}

  \begin{itemize}
  \item $G$ and $H$ are finitely generated free groups, rank $H > 1$
  \item $\phi, \psi: G \to H$ are homomorphisms
  \item Elements $u, v \in H$ are \emph{doubly-twisted conjugate} iff
    there is some $g \in G$ with 
\[ u = \phi(g) v \psi(g)^{-1}. \]
\item No algorithm exists to decide doubly-twisted conjugacy in free groups.
  \item $[u]$ denotes the doubly-twisted conjugacy class of $u$.

  \end{itemize}
}

\frame
{
  \frametitle{Motivation: Nielsen coincidence theory}
  If $f, g: X \to Y$ are maps on compact surfaces with boundary, their induced
  homomorphisms in $\pi_1$ are as in the setting above. 

  The \emph{coincidence set}:
\[ \Coin(f,g) = \{ x \in X \mid f(x) = g(x) \} \]

  \begin{itemize}
\item <2->
  Each coincidence $x$ has an associated doubly-twisted
  conjugacy class (the \emph{Nielsen class})
\item <3-> Two coincidences
  can be merged by a homotopy only if their Nielsen classes are
  the same.

\item <4->  The \emph{Nielsen number} $N(f,g)$ is the number of ``essential''
  Nielsen classes, and is a lower bound on the minimal number of
  coincidences when $f$ and $g$ are changed by homotopies. 

\item <5->  Computing $N(f,g)$ typically requires deciding
  doubly-twisted conjugacy relations.
\end{itemize}
}

\frame
{ \frametitle{Homomorphisms with remnant}

Let $G = \langle g_1, \dots, g_n \rangle$. We say that $\phi$
\emph{has remnant} if for every $i$, the word $\phi(g_i)$ has a
subword which does not cancel in any product like 
\[ \phi(g_j)^{\pm 1} \phi(g_i) \quad \text{or} \quad \phi(g_i)
\phi(g_j)^{\pm 1} \]
except $j=i$ with exponent $-1$.
\vfill
\pause
This is close to saying that $\{ \phi(g_1), \dots,
\phi(g_n) \}$ is Nielsen reduced.
\vfill
\pause
Very easy to check.
}

\frame
{ \frametitle{Remnants}
\framesubtitle{I said it was easy}
\[ \phi: \begin{array}{rcl}
a &\mapsto &a^3bab^{-2}\\
b &\mapsto &ba^4ba^{-2} 
\end{array} \]
This one \emph{has} remnant.

\vfill

The condition is easy to check, and is generically satisfied for long
words. (In fact for long words, we expect the remnant itself to be long.)
}

\section{The theorem}

\frame
{ \frametitle{Remnants and doubly twisted conjugacy classes}
Let $\phi^v$ be $\phi$, conjugated by $v$.

Let $*$ be the free product, and $\phi^v * \psi: G*G \to H$ the obvious homomorphism.
\pause
\vfill

\begin{thm*}
Let $u \neq v$. If $\phi^v * \psi$ has remnant, and the
remnants of $\phi^v(g_i)$ and $\psi(g_i)$ do not cancel in any product:
\[ \phi^v(g_i)v^{-1}u, \quad \psi(g_i)v^{-1}u, \quad
u^{-1}v\phi^v(g_i), \quad u^{-1}v\psi(g_i)\]
then
\[ [u] \neq [v]. \]
\end{thm*}

\pause

Note that these hypotheses are all likely to hold when $\phi$ and
$\psi$ are ``long''.
}

\frame
{ \frametitle{Density of non-conjugate pairs}
\begin{thm*}
If $\phi$ and $\psi$ have remnant, then random elements $u, v$ will
have $[u] \neq [v]$ with probability
\[ P \ge 1 - \frac1{e(r)} \]
where $e(r)$ is exponential in the remnant lengths of $\phi$ and $\psi$. 
\end{thm*}
\vfill
\pause

In fact, any random finite set of elements will have pairwise disjoint
twisted conjugacy classes with probability close to 1 as $\phi$ and
$\psi$ get long.
}

\section{Application}

\frame
{ \frametitle{Back to Nielsen theory}
This suggests a new approach to an old problem in Nielsen theory of
compact manifolds: 

Let $N(f,g)$ be the Nielsen number, $\MC(f,g)$ the minimal coincidence number.
\begin{itemize}
\item From definitions, $N(f,g) \le \MC(f,g)$.
\pause
\item (Wecken 1942, Schirmer, 1955) In dimension not equal to 2,
  $N(f,g) = \MC(f,g)$.
  \pause
\item (Jiang, 1984) An example where $N(f,g) \ne \MC(f,g)$ in
  dimension 2 (a surface with boundary).
\end{itemize}
\pause

Old question: What conditions on $f,g$ will guarantee $N(f,g) = \MC(f,g)$? Are these conditions commonly satisfied? 
\pause
\vfill
(Jiang, Guo, 1993): $N(f,\id) = \MC(f,\id)$ for homeomorphisms.
}

\frame
{\frametitle{A conjecture}
We conjecture that $N(f,g) = \MC(f,g)$ generically for surfaces with boundary. 
\vfill
\pause
Random group elements are typically in different classes, and so coincidences with
random Nielsen classes typically cannot be combined by homotopy. 
\vfill
\pause
So for random data, the number of Nielsen classes will be the
same as the number of coincidences. (But how random is it really?)
}

\frame
{\frametitle{Self-promotion}
The paper (soon) and an online twisted-conjugacy calculator are at 
\begin{itemize}
\item \url{http://www.messiah.edu/~cstaecker}
\end{itemize}
}
\end{document}
