\titleofpresentation{The elementary theory of Thompson's group $F$
is undecidable}

\presenter{Vladimir Tolstykh}  


\centerline{(joint work with Valery Bardakov)}

M.~Sapir \cite{tolstykh3} asked whether the elementary theory of Thompson's group
$F$ is decidable (see \cite{tolstykh1} for the background information
on Thompson's groups.) Since the group $F$
is riddled with copies of the restricted wreath product $\mathbf Z \,\operatorname{wr} \mathbf Z,$
he asked later whether there is a first-order definable (with
parameters) copy of $\mathbf Z \,\operatorname{wr} \mathbf Z$ in $F.$
Note that the elementary  theory of the group $\mathbf Z \,\operatorname{wr} \mathbf Z$ is hereditarily
undecidable, for the group $\mathbf Z \,\operatorname{wr} \mathbf Z$
first-order interprets the ring of integers $\mathbf Z$ \cite{tolstykh2}.

In our talk we shall explicitly construct a subgroup of $F$
which is isomorphic to $\mathbf Z \,\operatorname{wr} \mathbf Z$
and which is first-order definable in $F$ with certain
parameters, thereby showing
that the elementary theory of Thompson's group $F$ is hereditarily
undecidable.



\begin{ourbiblio}{99}  

\bibitem{tolstykh1} J.~Cannon, W.~Floyd, and W.~Parry, Introductory notes to
Richard Thompson's groups, Enseign. Math. 42 (1996) 215--256.

\bibitem{tolstykh2} G.~Noskov, The elementary theory of a finitely generated almost solvable group,
Izv. Akad. Nauk SSSR Ser. Mat. 47 (1983) 498--517.

\bibitem{tolstykh3}  Problem list of the workshop `Thompson's Group at 40 Years' (Jan. 11-14 2004,
American Institute of Mathematics, Palo Alto, California), available at
http://aimath.org/WWN/thompsonsgroup/thompsonsgroup.pdf
\end{ourbiblio}

\presenteraddress{Yeditepe University, \.Istanbul}  

\email{vtolstykh@yeditepe.edu.tr}

