## Gabriel Pallier’s notes of Chris Cashen’s Cambridge lecture 13-01-2017

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Notes from talk 29 at NPCW01, Cambridge, January 13th 2017. C. Cashen (UniversitÃ¤t Wien), joint work with J. Mackay.

\paragraph{Goal} Construct “boundaries of hyperbolic groups” for non hyperbolic groups.

This builds on the following (Charney-Sultan) : let ${X}$ be a ${\mathrm{CAT}(0)}$ space. Define its contracting boundary ${\partial_c X}$ as

$\displaystyle \partial_c X = \frac{\left\{ \text{contracting geodesic rays based at}\; o \right\}}{\text{bounded Hausdorff distance}}.$

Pros and cons : This ${\partial_c}$ is a QI invariant. However, for the Charney Sultan topology it is neither compact nor first countable (hence not metrizable) in general, and there is a loss of geometric intuition. On needs to find a better topology on this ${\partial_c X}$.

The topology of fellow-travelling quasi-geodesics on the contracting boundary of a f.g. group is a metrizable QI invariant.

\paragraph{Contractions} Let ${X}$ be a proper, geodesic metric space. For ${Z \subset X}$, define ${\pi_Z : X \rightarrow 2^Z}$ as

$\displaystyle x \mapsto \left\{ z \in Z : d(x,z) = d(x, Z) \right\}.$

${\pi_Z(x)}$ is never empty. There is no bound on its diameter. Let ${\rho}$ be a non-negative, non-decreasing sublinear function (i.e. ${\lim f(r)/r = 0}$).

Say ${Z}$ is ${\rho}$-contracting if for any ${x,y \in X}$,

$\displaystyle d(x,y) \leqslant d(x, Z) \implies \mathrm{diam}(\pi_Z(x) \cup \pi_Z(y)) \leqslant \rho (d(x,Z)).$

${X}$ is hyperbolic. Then ${Z}$ is contracting if and only if ${Z}$ is quasi-convex.

${Z}$ is strongly contracting if it is ${\rho}$-contracting for some bounded ${\rho}$.

Let ${\mu}$ be a function. ${Z}$ is ${\mu}$-Morse if for all ${L\geqslant 1}$, ${A \geqslant 0}$, and for any ${(L,A)}$ quasi-geodesic ${\gamma}$ with endpoints on ${Z}$, ${\gamma}$ lies in a ${\mu(L,A)}$-neighborhood of ${Z}$.

Given ${\rho, L, A}$ as above, there exists ${K,K'}$ and a ${\rho}$-contracting set ${Z}$ such that for any continuous ${(L,A)}$-quasigeodesic ray ${\gamma}$ starting on ${Z}$, either

• {${\gamma}$ is trapped : the subset of ${Z}$ that comes within distance ${K}$ of ${\gamma}$ is unbounded, and ${\gamma}$ stays in a ${K'}$ neighborhood of ${Z}$, or}
• {${\gamma}$ escapes : there exists a last point on ${Z}$ at distance ${K}$ from ${\gamma}$, after which ${\gamma}$ escapes “quickly”.}

In the second case, one can control the closest point on ${Z}$ from ${\gamma}$ by ${J(t) = \mathrm{dist}(\gamma(t), Z)}$ after escape point : this is smaller than ${\frac{4}{3} J + \mathrm{const}(\rho, L,A)}$.

\paragraph{The FQ topology} Define the following topology: let ${\eta}$ and ${\varrho}$ be equivalence classes of contracting geodesic rays. Then ${\eta}$ is close to ${\varrho}$ if geodesics in ${\eta}$ and ${\varrho}$ closely follow travel for a long time.

Carney-Sultan’s idea is to control the contraction constants. Define ${\partial_{c,\rho}^{FG} X = \left\{ \rho-\text{contracting rays} \right\} / \sim}$, and then take the direct limit on sublinear ${\rho}$

$\displaystyle \partial_c^{DL} X = \lim_{\rightarrow} \partial_{c,\rho}^{FG} X.$

${DL}$ is a QI invariant.

${\mathbb{Z}^2 \star \mathbb{Z} = \langle a,b,c \mid [a,b] \rangle}$.

The preceding example illustrates a non-convergence problem ; there is need for a new topology. For any ${\varrho \in \partial_c X}$, denote by ${\alpha^\varrho}$ a geodesic in ${\varrho}$.

For any ${\varrho}$ there exists a sublinear ${\rho^{\varrho}}$ so that all geodesics in ${\varrho}$ are ${\rho^{\varrho}}$-contracting.

One can define a neighborhood system ${\left\{ \mathcal{U}( \varrho, r ) \right\}_{\varrho \in \partial_c X, r \geqslant 1}}$ in a way such that a reasonable quasi geodesic in ${\eta}$ cannot escape from ${\alpha^\varrho}$ until after distance ${r}$. Denote by ${FQ}$ (follow-travelling quasigeodesic) the topology on ${\partial_c X}$ defined by : ${U}$ is open if ${\forall \varrho \in U, \exists r, \mathcal{U}(\varrho, r) \subset U}$.

Let ${X}$ be proper, geodesic metric space. Then ${\partial_c^{FQ} X}$ is Hausdorff and regular. The ${U(\varrho, n)}$ for ${n \in \mathbb{N}}$ form a neighborhood basis.

If ${G}$ is a finitely generated group, ${\partial_c^{FQ} G}$ is well-defined (since this is a QI invariant).

Let ${G}$ be a f.g. group. Then

1. {${\partial_c G}$ is non-empty ; }
2. {${\vert \partial_c G \vert = 2}$ iff ${G}$ is virtually ${\mathbb{Z}}$ ; }
3. {If ${\vert \partial_c G \vert = \infty}$, then ${G \curvearrowright \partial_c^{FQ} G}$ is minimal.}

Questions frrm the audience:

1. {Are there examples where this invariant ${FQ}$ is actually computable ?}
2. {Is there a prefered metric on ${\partial_c X}$ ?}

1. {Yes, some mixed free/direct products of ${\mathbb{Z}}$.}