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



Notes from talk 29 at NPCW01, Cambridge, January 13th 2017. C. Cashen (Universität 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}}.}
  2. {Working on it…}



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