Suppose that all hyperbolic groups are residually finite. The following statements follow: In relatively hyperbolic groups with peripheral structures consisting of finitely generated nilpotent subgroups, quasiconvex subgroups are separable; Geometrically finite subgroups of non-uniform lattices in rank one symmetric spaces are separable; Kleinian groups are subgroup separable. We also show that...

In this note we show that for any hyperbolic surface S, the number of geodesics of length bounded above by L in the mapping class group orbit of a fixed closed geodesic γ with a single double point is asymptotic to Ldim(Teichmuller space of S.). Since closed geodesics with one double point fall into a finite number of M≀⌈(S) orbits, we get the same asympotic estimate for the number of such geod...

Given a $p$-group $G$ and subgroup-closed class $\mathfrak{X}$, we associate with each $\mathfrak{X}$-subgroup $H$ certain quantities which count $\mathfrak{X}$-subgroups containing subject to further properties. We show in Theorem I that one of the said is always $\equiv 1 \pmod p$ if only same holds for others. In II supplement above result by focusing on normal III obtain sharpened version c...

Let G be a connected and reductive group over the algebraically closed field K. J-P. Serre has introduced the notion of a G-completely reducible subgroup H ≤ G. In this note, we give a notion of G-complete reducibility – G-cr for short – for Lie subalgebras of Lie(G), and we show that if the smooth subgroup H < G is G-cr, then Lie(H) is G-cr as well.

This paper investigates ways to enlarge the Hamiltonian subgroup Ham of the symplectomorphism group Symp(M) of the symplectic manifold (M,ω) to a group that both intersects every connected component of Symp(M) and characterizes symplectic bundles with fiber M and closed connection form. As a consequence, it is shown that bundles with closed connection form are stable under appropriate small per...

We define a strongly normalising proof-net calculus corresponding to the logic of strongly compact closed categories with biproducts. The calculus is a full and faithful representation of the free strongly compact closed category with biproducts on a given category with an involution. This syntax can be used to represent and reason about quantum processes.

It is shown that the commutator subgroup of the fundamental group of a smooth irreducible affine curve over an uncountable algebraically closed field k of positive characteristic is a profinite free group of rank equal to the cardinality of k.

Let S∞ denote the symmetric group on the natural numbers N. Then S∞ is a Polish group with the topology inherited from NN with the product topology and the discrete topology on N. Let d denote the least cardinality of a dominating family for NN and let c denote the continuum. Using theorems of Galvin, and Bergman and Shelah we prove that if G is any subgroup of S∞ that is closed in the above to...

Let M be a triangulable compact manifold. We prove that, among closed subgroups of Homeo0(M) (the identity component of the group of homeomorphisms of M), the subgroup consisting of volume preserving elements is maximal. AMS classification. 57S05 (57M60, 37E30).

Let G be a closed subgroup of S∞ and X be a Polish G-space. To each x ∈ X we associate an admissible set Ax and show how questions about X which involve Baire category, can be formalized in Ax.