Let M = 〈M, +, <, 0, {λ}λ∈D〉 be an ordered vector space over an ordered division ring D, and G = 〈G,⊕, eG〉 an n-dimensional group definable in M. We show that if G is definably compact and definably connected with respect to t-topology, then it is definably isomorphic to a ‘definable quotient group’ U/L, for some convex W -definable subgroup U of 〈Mn, +〉 and a lattice L of rank n. As two conseq...

We first show that co-amenability does not pass to subgroups, answering a question asked by Eymard in 1972. We then address address coamenability for von Neumann algebras, describing notably how it relates to the former. Co-amenable subgroups A subgroup H of a group G is called co-amenable in G if it has the following relative fixed point property: Every continuous affine G-action on a convex c...

We consider planar geometric models given by an explicit boundary of 0 en) algebraic curve segments of maximum degree d. We present an 0 en"dO(l) time algorithm to compute its convex hull and an 0 (nlog logn +K) . dO(l) time algorithms to compute various decompositions of an object, where K is the characteristic number of this object. Both operations, besides being solutions to interesting comp...

Let p be a polynomial in the non-commuting variables (a, x) = (a1, . . . , aga , x1, . . . , xgx). If p is convex in the variables x, then p has degree two in x and moreover, p has the form p = L + ΛΛ, where L has degree at most one in x and Λ is a (column) vector which is linear in x, so that ΛΛ is a both sum of squares and homogeneous of degree two. Of course the converse is true also. Furthe...

A Kleinian group Γ < Isom(H) is called convex cocompact if any orbit of Γ in H is quasiconvex or, equivalently, Γ acts cocompactly on the convex hull of its limit set in ∂H. Subgroup stability is a strong quasiconvexity condition in finitely generated groups which is intrinsic to the geometry of the ambient group and generalizes the classical quasiconvexity condition above. Importantly, it coin...

Let $$p \geqslant 3$$ be a prime integer and, for $$l 1$$ , let $$G \cong {{\mathbb {Z}}}_{p}^{l}$$ group of conformal automorphisms some closed Riemann surface S genus $$g 2$$ . By the Riemann–Hurwitz formula, either \leqslant g+1$$ or $$p=2\,g+1$$ If $$l=1$$ and then S/G is sphere with exactly three cone points if moreover 11$$ G unique p-Sylow subgroup $$\textrm{Aut}\hspace{0.55542pt}(S)$$ $...