Ruoyu LI

The fundamental group of a torus is well known, $Z^2$, whose fundamental group is $GL_2(Z)$. One way to see the geometry of
an automorphism is to make a mapping torus.

For a genus $g$ (where $g>2$) surface $\Sigma_g$, by the theorem of Dehn Nielsen, the mapping class group (the group of isotopy
classes of diffeomorphisms) of $\Sigma_g$ is isomorphic to a subgroup of index 2 of $Out(\pi_1(\Sigma_g))$. Thurston gave a theorem,
saying that its mapping class group is pseudo-anosov if and only if $\phi_1(\Sigma)\rtimes Z$ is Gromov hyperbolic.

For a free group $F_n$, Brinkman proved a theorem: an automorphism $\phi\in Aut(F_n)$ is aotoroidal if and only if $F_n\rtimes_{\phi} Z$ is Gromov hyperbolic.

My work is focused on the free product case, $G=G_1\ast\dots\ast G_p\ast F_k$, and consider $\phi\in Aut(G)$. I proved, by the "train
track technique", that if $G$ is non-elementary (k>3 or p+k>4) and if $\phi$ is fully irreducible and atroidal, then $G\rtimes_{\phi} Z$ is hyperbolic relative to the mapping torus of each $G_i$. From this, I also proved a theorem: if $\phi$ is atoroidal with
"central condition" (for all $i$, there exist $g_i\in G$ conjugating $\phi(G_i)$ to $G_i$, and there exist a non-trivial element of
$G_i\rtimes_{{\rm ad}_{g_i} \circ \phi|_{G_i}} Z$ that is central in $G_i\rtimes_{{\rm ad}_{g_i} \circ \phi|_{G_i}} Z$), then
$G\rtimes_{\phi} Z$ is hyperbolic relative to the mapping torus of each $G_i$.