Building on the algebraic framework developed by Hendricks, Manolescu, and Zemke, we introduce study a set of Floer-theoretic invariants aimed at detecting corks. Our obstruct extension given involution over any homology ball, rather than particular contractible manifold. Unlike previous approaches, do not utilize closed 4-manifold topology or contact topology. Instead, adapt formalism local eq...

According to Thurston’s stability theorem, every group of C diffeomorphisms of the closed interval is locally indicable (i.e., every finitely generated subgroup factors through Z). We show that, even for finitely generated groups, the converse of this statement is not true. More precisely, we show that the group F2 ⋉ Z , although locally indicable, does not embed into Diff1+(]0, 1[). (Here F2 i...

We study analogues of the notions from Lie theory of Levi subgroup and Levi decomposition, in the case of groups G definable in an ominimal expansion of a real closed field. With suitable definitions, we prove that G has a unique maximal ind-definable semisimple subgroup S, up to conjugacy, and that G = R ·S where R is the solvable radical of G. We also prove that any semisimple subalgebra of t...

In this note, we prove that a random extension of either the free group FN of rank N ě 3 or of the fundamental group of a closed, orientable surface Sg of genus g ě 2 is a hyperbolic group. Here, a random extension is one corresponding to a subgroup of either OutpFN q or ModpSgq generated by k independent random walks. Our main theorem is that a k–generated random subgroup of ModpSgq or OutpFN ...

We classify all locally finite joinings of a horospherical subgroup action on Γ\G when Γ is a Zariski dense geometrically finite subgroup of G = PSL2(R) or PSL2(C). This generalizes Ratner’s 1983 joining theorem for the case when Γ is a lattice in G. One of the main ingredients is equidistribution of non-closed horospherical orbits with respect to the Burger-Roblin measure which we prove in a g...

This paper deals with sufficiency conditions for irreducibility of certain induced modules. We also construct irreducible representations for a group G over a field K where the group G is a semidirect product of a normal abelian subgroup N and a subgroup H. The main results are proved with the assumption that charK does not divide |G| but there is no assumption made of K being algebraically clo...

We say that the width of an infinite subgroup H in G is n if there exists a collection of n essentially distinct conjugates of H such that the intersection of any two elements of the collection is infinite and n is maximal possible. We define the width of a finite subgroup to be 0. We prove that a quasiconvex subgroup of a negatively curved group has finite width. It follows that geometrically ...

Let G be a profinite group and let α be an automorphism of G. Then α is topologically intense if, for every closed subgroup H of G, there exists x ∈ G such that α(H) = xHx. Topologically intense automorphisms are automatically continuous, because they stabilize each open normal subgroup of the group on which they are defined. We denote by Intc(G) the group of topologically intense automorphisms...

We give a criterion in terms of the boundary for the existence of a proper cocompact action of a word-hyperbolic group on a CAT(0) cube complex. We describe applications towards lattices and hyperbolic 3-manifold groups. In particular, by combining the theory of special cube complexes, the surface subgroup result of Kahn-Markovic, and Agol’s criterion, we find that every subgroup separable clos...