Alessandro Sisto | Swiss Federal Institute of Technology (ETH) (original) (raw)

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Papers by Alessandro Sisto

We show that a relatively hyperbolic group quasiisometrically embeds in a product of finitely man... more We show that a relatively hyperbolic group quasiisometrically embeds in a product of finitely many trees if the peripheral subgroups do, and we provide an estimate on the minimal number of trees needed. Applying our result to the case of 3manifolds, we show that fundamental groups of closed 3-manifolds have asymptotic Assouad-Nagata dimension at most 8. To complement this result, we observe that fundamental groups of Haken 3manifolds with non-empty boundary have asymptotic dimension 2.

For relatively hyperbolic groups, we investigate conditions guaranteeing that the subgroup genera... more For relatively hyperbolic groups, we investigate conditions guaranteeing that the subgroup generated by two relatively quasiconvex subgroups Q 1 and Q 2 is relatively quasiconvex and isomorphic to Q 1 * Q1∩Q2 Q 2 . The main theorem extends results for quasiconvex subgroups of word-hyperbolic groups, and results for discrete subgroups of isometries of hyperbolic spaces.

The end compactification |Γ| of the locally finite graph Γ is the union of the graph and its ends... more The end compactification |Γ| of the locally finite graph Γ is the union of the graph and its ends, endowed with a suitable topology. We show that π1(|Γ|) embeds into a nonstandard free group with hyperfinitely many generators, i.e. an ultraproduct of finitely generated free groups, and that the embedding we construct factors through an embedding into an inverse limit of free groups. We also show how to recover the standard description of π1(|Γ|) given in . Finally, we give some applications of our result, including a short proof that certain loops in |Γ| are non-nullhomologous.

We show the equivalence of several characterizations of relative hyperbolicity for metric spaces,... more We show the equivalence of several characterizations of relative hyperbolicity for metric spaces, and obtain extra information about geodesics in a relatively hyperbolic space.

We study the bilipschitz equivalence type of tree-graded spaces, showing that asymptotic cones of... more We study the bilipschitz equivalence type of tree-graded spaces, showing that asymptotic cones of relatively hyperbolic groups (resp. asymptotic cones of groups containing a cut-point) only depend on the bilipschitz equivalence types of the pieces in the standard (resp. minimal) tree-graded structure. In particular, the asymptotic cones of many relatively hyperbolic groups do not depend on the scaling factor. We also describe the asymptotic cones as above "explicitly". Part of these results were obtained independently and simultaneously by D. Osin and M. Sapir in . 1 4 ALESSANDRO SISTO

We define a new notion of contracting element of a group and we show that contracting elements co... more We define a new notion of contracting element of a group and we show that contracting elements coincide with hyperbolic elements in relatively hyperbolic groups, pseudo-Anosovs in mapping class groups, rank one isometries in groups acting properly on proper CAT(0) spaces, elements acting hyperbolically on the Bass-Serre tree in graph manifold groups. We also define a related notion of weakly contracting element, and show that those coincide with hyperbolic elements in groups acting acylindrically on hyperbolic spaces and with iwips in Out(Fn)Out(F_n)Out(Fn), ngeq3n\geq 3ngeq3. We prove that any simple random walk in a non-elementary finitely generated subgroup containing a (weakly) contracting element ends up in a non-(weakly-)contracting element with exponentially decaying probability. Also, we show that each (weakly) contracting element is contained in a hyperbolically embedded elementary subgroup.

Arxiv preprint arXiv:1111.2499, Jan 1, 2011

We show that any group that is hyperbolic relative to virtually nilpotent subgroups, and does not... more We show that any group that is hyperbolic relative to virtually nilpotent subgroups, and does not admit certain splittings, contains a quasi-isometrically embedded copy of the hyperbolic plane. The specific embeddings we find remain quasiisometric embeddings when composed with the natural map from the Cayley graph to the coned-off graph, as well as when composed with the quotient map to "almost every" peripheral (Dehn) filling.

Arxiv preprint arXiv:1112.0263, Jan 1, 2011

We prove that the universal cover of any graph manifold quasiisometrically embeds into a product ... more We prove that the universal cover of any graph manifold quasiisometrically embeds into a product of three trees. In particular we show that the Assouad-Nagata dimension of the universal cover of any closed graph manifold is 3, proving a conjecture of Smirnov.

We describe the (minimal) tree-graded structure of asymptotic cones of non-geometric graph manifo... more We describe the (minimal) tree-graded structure of asymptotic cones of non-geometric graph manifold groups, and as a consequence we show that all said asymptotic cones are bilipschitz equivalent. Combining this with geometrization and other known results we obtain that all asymptotic cones of a given 3-manifold group are bilipschitz equivalent.

Iterated asymptotic cones have been used by Dru\c{t}u and Sapir to construct a group with uncount... more Iterated asymptotic cones have been used by Dru\c{t}u and Sapir to construct a group with uncountably many pairwise non-homeomorphic asymptotic cones. In this paper we define a class of metric spaces which display a wide range of behaviors with respect to iterated asymptotic cones, and we use those to construct examples within the class of groups. Namely, we will show that there exists a group whose iterated cones are pairwise non-homeomorphic, or periodically homeomorphic.

Geometriae Dedicata, Jan 1, 2009

Arxiv preprint arXiv:1010.4552, Jan 1, 2010

Arxiv preprint arXiv: …, Jan 1, 2010

Arxiv preprint arXiv:1107.2019, Jan 1, 2011

Arxiv preprint arXiv:1107.3494, Jan 1, 2011

Using ultrafilter techniques we show that in any partition of N into 2 cells there is one cell co... more Using ultrafilter techniques we show that in any partition of N into 2 cells there is one cell containing infinitely many exponential triples, i.e. triples of the kind a, b, a b (with a, b > 1). Also, we will show that any multiplicative IP * set is an "exponential IP set", the analogue of an IP set with respect to exponentiation.

We show that a relatively hyperbolic group quasiisometrically embeds in a product of finitely man... more We show that a relatively hyperbolic group quasiisometrically embeds in a product of finitely many trees if the peripheral subgroups do, and we provide an estimate on the minimal number of trees needed. Applying our result to the case of 3manifolds, we show that fundamental groups of closed 3-manifolds have asymptotic Assouad-Nagata dimension at most 8. To complement this result, we observe that fundamental groups of Haken 3manifolds with non-empty boundary have asymptotic dimension 2.

For relatively hyperbolic groups, we investigate conditions guaranteeing that the subgroup genera... more For relatively hyperbolic groups, we investigate conditions guaranteeing that the subgroup generated by two relatively quasiconvex subgroups Q 1 and Q 2 is relatively quasiconvex and isomorphic to Q 1 * Q1∩Q2 Q 2 . The main theorem extends results for quasiconvex subgroups of word-hyperbolic groups, and results for discrete subgroups of isometries of hyperbolic spaces.

The end compactification |Γ| of the locally finite graph Γ is the union of the graph and its ends... more The end compactification |Γ| of the locally finite graph Γ is the union of the graph and its ends, endowed with a suitable topology. We show that π1(|Γ|) embeds into a nonstandard free group with hyperfinitely many generators, i.e. an ultraproduct of finitely generated free groups, and that the embedding we construct factors through an embedding into an inverse limit of free groups. We also show how to recover the standard description of π1(|Γ|) given in . Finally, we give some applications of our result, including a short proof that certain loops in |Γ| are non-nullhomologous.

We show the equivalence of several characterizations of relative hyperbolicity for metric spaces,... more We show the equivalence of several characterizations of relative hyperbolicity for metric spaces, and obtain extra information about geodesics in a relatively hyperbolic space.

We study the bilipschitz equivalence type of tree-graded spaces, showing that asymptotic cones of... more We study the bilipschitz equivalence type of tree-graded spaces, showing that asymptotic cones of relatively hyperbolic groups (resp. asymptotic cones of groups containing a cut-point) only depend on the bilipschitz equivalence types of the pieces in the standard (resp. minimal) tree-graded structure. In particular, the asymptotic cones of many relatively hyperbolic groups do not depend on the scaling factor. We also describe the asymptotic cones as above "explicitly". Part of these results were obtained independently and simultaneously by D. Osin and M. Sapir in . 1 4 ALESSANDRO SISTO

We define a new notion of contracting element of a group and we show that contracting elements co... more We define a new notion of contracting element of a group and we show that contracting elements coincide with hyperbolic elements in relatively hyperbolic groups, pseudo-Anosovs in mapping class groups, rank one isometries in groups acting properly on proper CAT(0) spaces, elements acting hyperbolically on the Bass-Serre tree in graph manifold groups. We also define a related notion of weakly contracting element, and show that those coincide with hyperbolic elements in groups acting acylindrically on hyperbolic spaces and with iwips in Out(Fn)Out(F_n)Out(Fn), ngeq3n\geq 3ngeq3. We prove that any simple random walk in a non-elementary finitely generated subgroup containing a (weakly) contracting element ends up in a non-(weakly-)contracting element with exponentially decaying probability. Also, we show that each (weakly) contracting element is contained in a hyperbolically embedded elementary subgroup.

Arxiv preprint arXiv:1111.2499, Jan 1, 2011

We show that any group that is hyperbolic relative to virtually nilpotent subgroups, and does not... more We show that any group that is hyperbolic relative to virtually nilpotent subgroups, and does not admit certain splittings, contains a quasi-isometrically embedded copy of the hyperbolic plane. The specific embeddings we find remain quasiisometric embeddings when composed with the natural map from the Cayley graph to the coned-off graph, as well as when composed with the quotient map to "almost every" peripheral (Dehn) filling.

Arxiv preprint arXiv:1112.0263, Jan 1, 2011

We prove that the universal cover of any graph manifold quasiisometrically embeds into a product ... more We prove that the universal cover of any graph manifold quasiisometrically embeds into a product of three trees. In particular we show that the Assouad-Nagata dimension of the universal cover of any closed graph manifold is 3, proving a conjecture of Smirnov.

We describe the (minimal) tree-graded structure of asymptotic cones of non-geometric graph manifo... more We describe the (minimal) tree-graded structure of asymptotic cones of non-geometric graph manifold groups, and as a consequence we show that all said asymptotic cones are bilipschitz equivalent. Combining this with geometrization and other known results we obtain that all asymptotic cones of a given 3-manifold group are bilipschitz equivalent.

Iterated asymptotic cones have been used by Dru\c{t}u and Sapir to construct a group with uncount... more Iterated asymptotic cones have been used by Dru\c{t}u and Sapir to construct a group with uncountably many pairwise non-homeomorphic asymptotic cones. In this paper we define a class of metric spaces which display a wide range of behaviors with respect to iterated asymptotic cones, and we use those to construct examples within the class of groups. Namely, we will show that there exists a group whose iterated cones are pairwise non-homeomorphic, or periodically homeomorphic.

Geometriae Dedicata, Jan 1, 2009

Arxiv preprint arXiv:1010.4552, Jan 1, 2010

Arxiv preprint arXiv: …, Jan 1, 2010

Arxiv preprint arXiv:1107.2019, Jan 1, 2011

Arxiv preprint arXiv:1107.3494, Jan 1, 2011

Using ultrafilter techniques we show that in any partition of N into 2 cells there is one cell co... more Using ultrafilter techniques we show that in any partition of N into 2 cells there is one cell containing infinitely many exponential triples, i.e. triples of the kind a, b, a b (with a, b > 1). Also, we will show that any multiplicative IP * set is an "exponential IP set", the analogue of an IP set with respect to exponentiation.