448 research outputs found
Finite index subgroups without unique product in graphical small cancellation groups
We construct torsion-free hyperbolic groups without unique product whose
subgroups up to some given finite index are themselves non-unique product
groups. This is achieved by generalising a construction of Comerford to
graphical small cancellation presentations, showing that for every subgroup
of a graphical small cancellation group there exists a free group such that
admits a graphical small cancellation presentation.Comment: 8 pages, 1 figur
Characterizations of Morse quasi-geodesics via superlinear divergence and sublinear contraction
We introduce and begin a systematic study of sublinearly contracting
projections. We give two characterizations of Morse quasi-geodesics in an
arbitrary geodesic metric space. One is that they are sublinearly contracting;
the other is that they have completely superlinear divergence. We give a
further characterization of sublinearly contracting projections in terms of
projections of geodesic segments.Comment: 24 pages, 5 figures. v2: 22 pages, 5 figures. Correction in proof of
Thm 7.1. Proof of Prop 4.2 revised for improved clarity. Other minor changes
per referee comments. To appear in Documenta Mathematic
Infinitely presented graphical small cancellation groups are acylindrically hyperbolic
We prove that infinitely presented graphical Gr(7) small cancellation groups are acylindrically hyperbolic. In particular, infinitely presented classical C(7)-groups and, hence, classical C'(1/6)-groups are acylindrically hyperbolic. We also prove the analogous statements for the larger class of graphical small cancellation presentations over free products. We construct infinitely presented classical C'(1/6)-groups that provide new examples of divergence functions of groups.ISSN:0373-0956ISSN:1777-531
Negative curvature in graphical small cancellation groups
We use the interplay between combinatorial and coarse geometric versions of
negative curvature to investigate the geometry of infinitely presented
graphical small cancellation groups. In particular, we characterize
their 'contracting geodesics', which should be thought of as the geodesics that
behave hyperbolically.
We show that every degree of contraction can be achieved by a geodesic in a
finitely generated group. We construct the first example of a finitely
generated group containing an element that is strongly contracting with
respect to one finite generating set of and not strongly contracting with
respect to another. In the case of classical small cancellation
groups we give complete characterizations of geodesics that are Morse and that
are strongly contracting.
We show that many graphical small cancellation groups contain
strongly contracting elements and, in particular, are growth tight. We
construct uncountably many quasi-isometry classes of finitely generated,
torsion-free groups in which every maximal cyclic subgroup is hyperbolically
embedded. These are the first examples of this kind that are not subgroups of
hyperbolic groups.
In the course of our analysis we show that if the defining graph of a
graphical small cancellation group has finite components, then the
elements of the group have translation lengths that are rational and bounded
away from zero.Comment: 40 pages, 14 figures, v2: improved introduction, updated statement of
Theorem 4.4, v3: new title (previously: "Contracting geodesics in infinitely
presented graphical small cancellation groups"), minor changes, to appear in
Groups, Geometry, and Dynamic
Random triangular Burnside groups
We introduce a model for random groups in varieties of -periodic groups as
-periodic quotients of triangular random groups. We show that for an
explicit , for densities
and for large enough, the model produces
\emph{infinite} -periodic groups. As an application, we obtain, for every
fixed large enough , for every an infinite -periodic
group with fixed points for all isometric actions on -spaces. Our main
contribution is to show that certain random triangular groups are uniformly
acylindrically hyperbolic.Comment: v1: 9 pages, 1 figure; v2: 14 pages, 1 figure. Expanded exposition,
final versio
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