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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 be a space. Define its contracting boundary as

Pros and cons : This 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 .

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

\paragraph{Contractions} Let be a proper, geodesic metric space. For , define as

is never empty. There is no bound on its diameter. Let be a non-negative, non-decreasing sublinear function (i.e. ).

Say is -contracting if for any ,

is hyperbolic. Then is contracting if and only if is quasi-convex.

is strongly contracting if it is -contracting for some bounded .

Let be a function. is -Morse if for all , , and for any quasi-geodesic with endpoints on , lies in a -neighborhood of .

Given as above, there exists and a -contracting set such that for any continuous -quasigeodesic ray starting on , either

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

In the second case, one can control the closest point on from by after escape point : this is smaller than .

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

Carney-Sultan’s idea is to control the contraction constants. Define , and then take the direct limit on sublinear

is a QI invariant.

.

The preceding example illustrates a non-convergence problem ; there is need for a new topology. For any , denote by a geodesic in .

For any there exists a sublinear so that all geodesics in are -contracting.

One can define a neighborhood system in a way such that a reasonable quasi geodesic in cannot escape from until after distance . Denote by (follow-travelling quasigeodesic) the topology on defined by : is open if .

Let be proper, geodesic metric space. Then is Hausdorff and regular. The for form a neighborhood basis.

If is a finitely generated group, is well-defined (since this is a QI invariant).

Let be a f.g. group. Then

- { is non-empty ; }
- { iff is virtually ; }
- {If , then is minimal.}

Questions frrm the audience:

- {Are there examples where this invariant is actually computable ?}
- {Is there a prefered metric on ?}

Answers:

- {Yes, some mixed free/direct products of .}
- {Working on it…}

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