Continuously many quasiisometry classes of 2-generator groups

We construct continuously many quasiisometry classes of torsion-free 2-generator small cancellation groups. Received: January 5, 1997     

Bestimmung des Schwefels in Schwefelmetallen

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Markoff triples and quasifuchsian groups

We study the global behaviour of trees of Markoff triples overthe complex numbers. We relate this to the space of type-preservingrepresentations of the punctured torus group into SL(2,C). Inparticular, we explore which Markoff triples correspond to quasifuchsianrepresentations. We derive a variation of McShane's identityfor quasifuchsian groups. In the case of non-discrete representations,we attempt to relate the asymptotic behaviour of Markoff triplesto the realisability of laminations in hyperbolic 3-space. Wealso consider how some of these issues might be related formore general surfaces. 1991 Mathematics Subject Classification:57M50.     

Tight geodesics in the curve complex

The curve graph, , associated to a compact surface Σ is the 1-skeleton of the curve complex defined by Harvey. Masur and Minsky showed that this graph is hyperbolic and defined the notion of a tight geodesic therein. We prove some finiteness results for such geodesics. For example, we show that a slice of the union of tight geodesics between any pair of points has cardinality bounded purely in terms of the topological type of Σ. We deduce some consequences for the action of the mapping class group on . In particular, we show that it satisfies an acylindricity condition, and that the stable lengths of pseudoanosov elements are rational with bounded denominator. Mathematics Subject Classification (2000) 20F32     

Peripheral splittings of groups

We define the notion of a ``peripheral splitting' of a group. This is essentially a representation of the group as the fundamental group of a bipartite graph of groups, where all the vertex groups of one colour are held fixed--the ``peripheral subgroups'. We develop the theory of such splittings and prove an accessibility result. The theory mainly applies to relatively hyperbolic groups with connected boundary, where the peripheral subgroups are precisely the maximal parabolic subgroups. We show that if such a group admits a non-trivial peripheral splitting, then its boundary has a global cut point. Moreover, the non-peripheral vertex groups of such a splitting are themselves relatively hyperbolic. These results, together with results from elsewhere, show that under modest constraints on the peripheral subgroups, the boundary of a relatively hyperbolic group is locally connected if it is connected. In retrospect, one further deduces that the set of global cut points in such a boundary has a simplicial treelike structure.

     

Connectedness properties of limit sets

We study convergence group actions on continua, and give a criterion which ensures that every global cut point is a parabolic fixed point. We apply this result to the case of boundaries of relatively hyperbolic groups, and consider implications for connectedness properties of such spaces.

     

Embedding median algebras in products of trees

We show that a metric median algebra satisfying certain conditions admits a bilipschitz embedding into a finite product of $\mathbb{R }$ -trees. This gives rise to a characterisation of closed connected subalgebras of finite products of complete $\mathbb{R }$ -trees up to bilipschitz equivalence. Spaces of this sort arise as asymptotic cones of coarse median spaces. This applies to a large class of finitely generated groups, via their Cayley graphs. We show that such groups satisfy the rapid decay property. We also recover the result of Behrstock, Dru?u and Sapir, that the asymptotic cone of the mapping class group embeds in a finite product of $\mathbb{R }$ -trees.