A note on automatic continuity

We show that if G is a non-elementary torsion-free word hyperbolic group then there exists another word hyperbolic group G∗, such that G is a subgroup of G∗ but G is not quasiconvex in G∗ . We also prove that any non-elementary subgroup of a torsion-free word hyperbolic group G contains a free group of rank two which is malnormal and quasiconvex in G.

Preface 1. Cayley's theorems 2. Groups generated by reflections 3. Groups acting on trees 4. Baumslag-Solitar groups 5. Words and Dehn's word problem 6. A finitely-generated, infinite, Torsion group 7. Regular languages and normal forms 8. The Lamplighter group 9. The geometry of infinite groups 10. Thompson's group 11. The large-scale geometry of groups Bibliography Index.

We present a general framework for automatic continuity results for groups of isometries of metric spaces. In particular, we prove automatic continuity property for the groups of isometries of the Urysohn space and the Urysohn sphere, i.e. that any homomorphism from either of these groups into a separable group is continuous. This answers a question of Ben Yaacov, Berenstein and Melleray. As a consequence, we get that the group of isometries of the Urysohn space has unique Polish group topology and the group of isometries of the Urysohn sphere has unique separable group topology. Moreover, as an application of our framework we obtain new proofs of the automatic continuity property for the group $\text{Aut}([0,1],\unicode[STIX]{x1D706})$, due to Ben Yaacov, Berenstein and Melleray and for the unitary group of the infinite-dimensional separable Hilbert space, due to Tsankov.

Let G be a group and G' its commutator subgroup. Commutator length (cl) and stable commutator length (scl) are naturally defined concepts for elements of G'. We study cl and scl for two classes of groups. First, we compute scl in generalized Thompson's groups and their central extensions. As a consequence, we find examples of finitely presented groups in which scl takes irrational (in fact, transcendental) values. Second, we study large scale geometry of the Cayley graph of a commutator subgroup with respect to the canonical generating set of all commutators. When G is a non-elementary hyperbolic group, we prove that, for any n, there exists a quasi-isometrically embedded, dimension n integral lattice in this graph. Thus this graph is not hyperbolic, has infinite asymptotic dimension, and has only one end. For a general finitely presented group, we show that this graph is large scale simply connected.

The flat-rank of a totally disconnected, locally compact group G is an integer, which is an invariant of G as a topological group. We generalize the concept of hyperbolic groups to the topological context and show that a totally disconnected, locally compact, hyperbolic group has flat-rank at most 1. It follows that the simple totally disconnected locally compact groups constructed by Paulin and Haglund have flat-rank at most 1.

A study on weakly Lindelöf (para)topological groups with a q-point and m([formula omitted])-factorizability

We explore the topological full group 〚 G 〛 $\llbracket G\rrbracket $ of an essentially principal étale groupoid G on a Cantor set. When G is minimal, we show that 〚 G 〛 $\llbracket G\rrbracket $ (and its certain normal subgroup) is a complete invariant for the isomorphism class of the étale groupoid G. Furthermore, when G is either almost finite or purely infinite, the commutator subgroup D ( 〚 G 〛 ) $D(\llbracket G\rrbracket )$ is shown to be simple. The étale groupoid G arising from a one-sided irreducible shift of finite type is a typical example of a purely infinite minimal groupoid. For such G, 〚 G 〛 $\llbracket G\rrbracket $ is thought of as a generalization of the Higman–Thompson group. We prove that 〚 G 〛 $\llbracket G\rrbracket $ is of type F ∞, and so in particular it is finitely presented. This gives us a new infinite family of finitely presented infinite simple groups. Also, the abelianization of 〚 G 〛 $\llbracket G\rrbracket $ is calculated and described in terms of the homology groups of G.

Bounded normal generation and invariant automatic continuity

We show that all amenable, minimal actions of a large class of nonamenable countable groups on compact metric spaces have dynamical comparison. This class includes all nonamenable hyperbolic groups, many HNN-extensions, nonamenable Baumslag–Solitar groups, a large class of amalgamated free products, lattices in many Lie groups, A ~ 2 {\widetilde{A}_{2}} -groups, as well as direct products of the above with arbitrary countable groups. As a consequence, crossed products by amenable, minimal and topologically free actions of such groups on compact metric spaces are Kirchberg algebras in the UCT class, and are therefore classified by K-theory.

A group G is called a CSA-group if all its maximal Abelian subgroups are malnormal; that is, Mx ∩ M = 1 for every maximal Abelian subgroup M and x ∈ G − M. The class of CSA-groups contains all torsion-free hyperbolic groups and all groups freely acting on λ-trees. We describe conditions under which HNN-extensions and amalgamated products of CSA-groups are again CSA. One-relator CSA-groups are characterised as follows: a torsion-free one-relator group is CSA if and only if it does not contain F2 × Z or one of the nonabelian metabelian Baumslag-Solitar groups B1, n = 〈x, y | yxy−1 = xn〉, n ∈ Z ∂ {0, 1}; a one-relator group with torsion is CSA if and only if it does not contain the infinite dihedral group.

We prove that there exists a countable metrizable topological group G G such that every countable metrizable group is isomorphic to a quotient of G G . The completion H H of G G is a Polish group such that every Polish group is isomorphic to a quotient of H H .

The Banach manifold structure of the space of metrics on noncompact manifolds