75,258 research outputs found
Invariant random subgroups of groups acting on hyperbolic spaces
Suppose that a group acts non-elementarily on a hyperbolic space and
does not fix any point of . A subgroup is said to be
geometrically dense in if the limit sets of and coincide and
does not fix any point of . We prove that every invariant random
subgroup of is either geometrically dense or contained in the elliptic
radical (i.e., the maximal normal elliptic subgroup of ). In particular,
every ergodic measure preserving action of an acylindrically hyperbolic group
on a Borel probability space either has finite stabilizers
-almost surely or otherwise the stabilizers are very large (in particular,
acylindrically hyperbolic) -almost surely.Comment: Minor corrections. To appear in Proc. AM
Polygonal words in free groups
A longstanding question of Gromov asks whether every one-ended
word-hyperbolic group contains a subgroup isomorphic to the fundamental group
of a closed hyperbolic surface. An infinite family of word-hyperbolic groups
can be obtained by taking doubles of free groups amalgamated along words that
are not proper powers. We define a set of polygonal words in a free group of
finite rank, and prove that polygonality of the amalgamating word guarantees
that the associated square complex virtually contains a -injective
closed surface. We provide many concrete examples of classes of polygonal
words. For instance, in the case when the rank is 2, we establish polygonality
of words without an isolated generator, and also of almost all simple height 1
words, including Baumslag--Solitar relator for .Comment: 23 pages, 8 figure
Two-Generator Free Kleinian Groups and Hyperbolic Displacements
The Theorem, proved by Culler and Shalen, states that every point in
the hyperbolic 3-space is moved a distance at least by one of the
non-commuting isometries or provided that and
generate a torsion-free, discrete group which is not co-compact and contains no
parabolic. This theorem lies in the foundation of many techniques that provide
lower estimates for the volumes of orientable, closed hyperbolic 3-manifolds
whose fundamental group has no 2-generator subgroup of finite index and, as a
consequence, gives insights into the topological properties of these manifolds.
In this paper, we show that every point in the hyperbolic 3-space is moved a
distance at least by one of the isometries in
when and satisfy the conditions given in
the Theorem.Comment: 43 Pages. 2 figures. Almost completely rewritten in line with the
referee's recommendations. By Lemma 4.10, suggested by the referee, the
proofs of many lemmas are substantially shortened. The main theorem, Theorem
5.1, is reproved with a more geometric approac
Quasi-hyperbolic planes in relatively hyperbolic groups
We show that any group that is hyperbolic relative to virtually nilpotent
subgroups, and does not admit peripheral splittings, contains a
quasi-isometrically embedded copy of the hyperbolic plane. In natural
situations, the specific embeddings we find remain quasi-isometric embeddings
when composed with the inclusion 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.
We apply our theorem to study the same question for fundamental groups of
3-manifolds.
The key idea is to study quantitative geometric properties of the boundaries
of relatively hyperbolic groups, such as linear connectedness. In particular,
we prove a new existence result for quasi-arcs that avoid obstacles.Comment: v1: 32 pages, 4 figures. v2: 38 pages, 4 figures. v3: 44 pages, 4
figures. An application (Theorem 1.2) is weakened as there was an error in
its proof in section 7, all other changes minor, improved expositio
Fuchsian groups, coverings of Riemann surfaces, subgroup growth, random quotients and random walks
AbstractFuchsian groups (acting as isometries of the hyperbolic plane) occur naturally in geometry, combinatorial group theory, and other contexts. We use character-theoretic and probabilistic methods to study the spaces of homomorphisms from Fuchsian groups to symmetric groups. We obtain a wide variety of applications, ranging from counting branched coverings of Riemann surfaces, to subgroup growth and random finite quotients of Fuchsian groups, as well as random walks on symmetric groups. In particular, we show that, in some sense, almost all homomorphisms from a Fuchsian group to alternating groups An are surjective, and this implies Higman's conjecture that every Fuchsian group surjects onto all large enough alternating groups. As a very special case, we obtain a random Hurwitz generation of An, namely random generation by two elements of orders 2 and 3 whose product has order 7. We also establish the analogue of Higman's conjecture for symmetric groups. We apply these results to branched coverings of Riemann surfaces, showing that under some assumptions on the ramification types, their monodromy group is almost always Sn or An. Another application concerns subgroup growth. We show that a Fuchsian group Γ has (n!)μ+o(1) index n subgroups, where μ is the measure of Γ, and derive similar estimates for so-called Eisenstein numbers of coverings of Riemann surfaces. A final application concerns random walks on alternating and symmetric groups. We give necessary and sufficient conditions for a collection of ‘almost homogeneous’ conjugacy classes in An to have product equal to An almost uniformly pointwise. Our methods involve some new asymptotic results for degrees and values of irreducible characters of symmetric groups
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