<|endoftext|>anski:1998fo; @Hruska:2010iw]. For our purpose we will use the following one.
\[def: relatively hyperbolic\] Let $G$ be a group and $\{ P_1, \dots, P_m\}$ be a collection of subgroups of $G$. We say that $G$ is *hyperbolic relative to* $\{P_1, \dots, P_m\}$ if there exist a proper geodesic hyperbolic space $X$ and a collection $\mathcal Y$ of pairwise disjoint open horoballs satisfying the following properties.
1. $G$ acts properly by isometries on $X$ and $\mathcal Y$ is $G$-invariant.
2. If $U$ stands for the union of the horoballs of $\mathcal Y$ then $G$ acts co-compactly on $X \setminus U$.
3. $\{P_1, \dots, P_m\}$ is a set of representatives of the $G$-orbits of $\set{\stab Y}{Y \in \mathcal Y}$.
The action of $G$ on the space $X$ given by is not acylindrical. Indeed the subgroups $P_j$ can be parabolic. This cannot happen with an acylindrical action [@Bowditch:2008bj Lemma 2.2]. More generally, the elementary subgroups of $G$ are exactly the virtually cyclic subgroups of $G$ and the ones which are conjugated to a subgroup of some $P_j$. As in the case of groups with an acylindrical action, one can prove that $\inj[X]G$ is positive whereas $\nu(G,X)$ and $A(G,X)$ are finite. Proceeding as in we get the following result.
\[res: SC - partial periodic quotient - rel hyp case\] Let $G$ be a group and $\{P_1, \dots, P_m\}$ be a collection of subgroups of $G$ such that $G$ is hyperbolic relatively to $\{P_1, \dots, P_m\}$. Assume that there are only finitely many isomorphism classes of finite subgroups with dihedral shape. There exist $p,N_1\in \N$ such that every integer $n \geq N_1$ multiple of $p$, there exists a quotient $Q$ of $G$ with the following properties.
1. if $E$ is a finite subgroup of $G$ or conjugated to some $P_j$, then the projection $G \onto Q$ induces an isomorphism from $E$ onto its image;
2. for every element $g \in Q$, either $g^n=1$ or $g$ is the image a non-loxodromic element of $G$;
3. there are infinitely many elements in $Q$ which do not belong to the image of an elementary non-loxodromic subgroup of $G$.
#### Mapping class groups.
Let $\Sigma$ be a compact surface of genus $g$ with $p$ boundary components. In the rest of this paragraph we assume that its complexity $3g+p -3$ is larger than $1$. The *mapping class group* $\mcg \Sigma$ of $\Sigma$ is the group of orientation preserving self homeomorphisms of $\Sigma$ defined up to homotopy. A mapping class $f \in \mcg \Sigma$ is
1. *periodic*, if it has finite order;
2. *reducible*, if it permutes a collection of essential non-peripheral curves (up to isotopy);
3. *pseudo-Anosov*, if there exists an homeomorphism in the class of $f$ that preserves a pair of transverse foliations and rescale these foliations in an appropriate way.
It follows from Thurston’s work that any element of $\mcg \Sigma$ falls into one these three categories [@Thurston:1988fa Theorem 4]. The *complex of curves* $X$ is a simplicial complex associated to $\Sigma$. It has been first introduced by Harvey [@Harvey:1981tg]. A $k$-simplex of $X$ is a collection of $k+1$ homotopy classes of curves of $\Sigma$ that can be disjointly realized. Masur and Minsky proved that this new space is hyperbolic [@Masur:1999hc]. By construction, $X$ is endowed with an action by isometries of $\mcg \Sigma$. Moreover Bowditch showed that this action is acylindrical [@Bowditch:2008bj Theorem 1.3]. This action provides an other characterization of the elements of $\mcg \Sigma$. An element of $\mcg \Sigma$ is periodic or reducible (pseudo-Anosov) if and