## BI: Relatively hyperbolic groups, part 1

There are essentially three families of characterizations of relatively hyperbolic groups, corresponding to three different point of views. I’ll present two of them in this post, the ones I’m more familiar with, while in part 2 I’ll write about the other one and about examples of relatively hyperbolic groups. Let’s get started!

It might be useful to bear in mind that a group $G$ can be hyperbolic relative to a collection of subgroups, called peripheral subgroups.

1) Relatively hyperbolic groups coarsely look like tree-graded spaces, the pieces being left cosets of peripheral subgroups.
This point of view has been invented by Cornelia Druţu and Mark Sapir. If you’re not familiar with tree-graded spaces you might wish to take a look at this post. In any case, here’s a picture: Just to make me feel better, here is a possible way of formalizing the idea of “coarsely looking like a tree-graded space”, which you can just ignore if you don’t know what asymptotic cones are.

Theorem/definition: A group $G$ is hyperbolic relative to  its subgroups $P_1,\dots, P_n$ if and only if all its asymptotic cones are tree-graded, with pieces given by ultralimits of left cosets of $P_1,\dots,P_n$.

Once again, $P_1,\dots,P_n$ are called peripheral subgroups.

Let me now explain the title. The idea is that is something is true for tree-graded spaces then some coarse version is true for relatively hyperbolic groups. For example, given a point $x$ and a piece $P$ in a tree-graded space, there exists a point usually denote by $\pi_P(x)$ on $P$ such that all geodesics from $x$ to $P$ contain $\pi_P(x)$. Well, in relatively hyperbolic groups, when $P$ is a (left coset of a) peripheral subgroup, the property that $\pi_P(x)$ has is that all geodesics from $x$ to $P$ pass close to $\pi_P(x)$, i.e. they intersect a ball of a certain radius around $\pi_P(x)$. I’ll get back on these projections soon…
Finally, let me show how easy facts about tree-graded spaces sometimes translate into interesting facts about relatively hyperbolic groups.

Lemma: peripheral subgroups are quasi-convex, and so in particular they are undistorted.

Recall that a subspace is quasi-convex if there exists a constant $C$ such that geodesics connecting points on the subspace stay $C -$close to the subspace (undistorted means that the inclusion of the corresponding Cayley graphs is a quasi-isometric embedding). Why is this true? Well, pieces in tree-graded spaces are convex (you see this, right?), and the coarse version of this property is being quasi-convex.
Similarly, if $P$ is a peripheral subgroup and $P\neq gPg^{-1}$ then $P \cap gP g^{-1}$ is finite (i.e., $P$ is almost malnormal). Exercise: convince yourself of this fact!

2) point of view: a relatively hyperbolic group acts properly on a hyperbolic space, peripheral subgroups being maximal parabolic subgroups.
This point of view has been invented by Gromov and adopted by, among others, Bowditch, Groves and Manning, Dahmani…
Recall that an isometry of the (Gromov-)hyperbolic space $X$ is called parabolic if the induced homeomorphism  of the boundary $\partial X$ has exactly one fixed point. Rather than giving you a statement like “the group $G$ is relatively hyperbolic if and only if there exists such and such action on a hyperbolic space $X(G)$” I prefer to give you the construction of $X (G)$.

In order to do so, let’s define what we will call Bowditch horoballs. Let $\Gamma$ be a metric graph, say with edges of length 1. ( $\Gamma$ will be a Cayley graph of a peripheral subgroup.) Let $\latex V$ be the set of vertices of $\Gamma$, and let the Bowditch horoball $H(\Gamma)$ be the metric graph with vertices $V\times \mathbb{N}$ and the “obvious” horizontal and vertical edges. The length of the vertical edges is always 1, while the length of horizontal edges at height $n$ is $1/2^{-n}$.
It is easy to see that geodesics go straight up, then travel along a geodesic of length at most 4 in the path metric of a certain level, and then go straight down (subsequences of this sequence of “moves” are also allowed). In fact, for example, the red path is clearly not a geodesic as the blue path is shorter: Exercise: show that for $x,y\in \Gamma$, seen as a subspace of $H(\Gamma)$, $d_{H(\Gamma)}$ is up to finite additive error $2\log_2(d_\Gamma(x,y)+1)$.
As it turns out, and I would really like to have a deeper understanding of this fact, Bowditch horoballs are always hyperbolic spaces (!) and their boundaries consist of a single point, corresponding to geodesic rays going straight up.

Ok, back to relatively hyperbolic groups. Let $P_1,\dots, P_n be groups (all of them finitely generated). Fix a finite system of generators for $G$ that restricts to a system of generators for each $P_i$. To avoid repeating “Cayley graph” over and over again, we’ll silently identify $G, P_i, \dots$ with the corresponding Cayley graphs in the system of generators we fixed.
Let $X(G)$ be obtained gluing to each left coset of $P_i$ a copy of $H(P_i)$. (Indeed, we should write $X(G,P_1,\dots,P_n)$.)

Theorem/definition: $G$ is hyperbolic relative to $P_1,\dots, P_n$  if and only if $X(G)$ is hyperbolic.

There is a natural proper action of $G$ on $X(G)$. Peripheral subgroups and their conjugates act parabolically (one can see the boundary of the glued-in Bowditch horoballs in $\partial X(G)$), and are maximal with this property.

The nice feature of this characterization, to put it bluntly, is that one can prove stuff just drawing a few trees.
In fact, finite configurations of geodesics in hyperbolic spaces can be approximated with trees. For example, triangles can be approximated by tripods: Another good thing to know is that the cosets of $P_i$, as subsets of $X(G)$, are coarsely horospheres, in the sense that they coincide up to finite Hausdorff distance with level sets of Busemann functions. Here is how to estimate Busemann functions. Let $c\in\partial X(G)$ and $x,w\in X(G)$. Suppose that the situation looks as follows: Then the Busemann function $\beta_c(x,w)$ is approximately $d(x,p)-d(x,w)$ (i.e. up to finite error).
This is quite useful in computations, examples will hopefully show up in other posts.

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