Let X be a locally finite tree, and let G = Aut(X). Then G is a locally compact group. We show that if X has more than one end, and if G contains a discrete subgroup Γ such that the quotient graph of groups Γ\\X is infinite but has finite covolume, then G contains a non-uniform lattice, that is, a discrete subgroup Λ such that Λ\G is not compact, yet has a finite G-invariant measure. 0. Notatio...

Let $X$ be a (smooth) compact complex surface. We show that the torsion subgroup of biholomorphic automorphism group $\operatorname{Aut}(X)$ is virtually nilpotent. Moreover, we study Tits alternative and virtual derived length solvable subgroups $\operatorname{Aut}(X)$.

Let X be a compact metrizable abelian group and u = {un} be a sequence in its dual X. Set su(X) = {x : (un, x) → 1} and T H 0 = {(zn) ∈ T ∞ : zn → 1}. Let G be a subgroup of X. We prove that G = su(X) for some u iff it can be represented as some dually closed subgroup Gu of ClXG × T H 0 . In particular, su(X) is polishable. Let u = {un} be a T -sequence. Denote by (X̂,u) the group X ∧ equipped w...

in this paper, the concept of fuzzy convex subgroup (resp. fuzzy convex lattice-ordered subgroup) of an ordered group (resp. lattice-ordered group) is introduced and some properties, characterizations and related results are given. also, the fuzzy convex subgroup (resp. fuzzy convex lattice-ordered subgroup) generated by a fuzzy subgroup (resp. fuzzy subsemigroup) is characterized. furthermore,...

A sequence {un}n∈ω in abstract additively-written Abelian group G is called a T -sequence if there is a Hausdorff group topology on G relative to which limn un = 0. We say that a subgroupH of an infinite compact Abelian groupX is T -characterized if there is a T -sequence u = {un} in the dual group of X , such that H = {x ∈ X : (un, x) → 1}. We show that a closed subgroupH ofX is T -characteriz...

Homework Policy. Please read through all the problems. I will be happy to discuss the solutions during office hours. Problems. Problem 1. For a finite type, separated k-scheme X, recall the alternative definition of the subgroup Ratl(X) ⊂ Zl(X). It equals the subgroup generated by all pushforward classes, i∗[W0]− i∗[W∞], for all k-schemes W of pure dimension l + 1, and for all proper morphisms ...

We classify pairs (X,G) consisting of a complex K3 surface X and finite group G≤Aut(X) such that the subgroup G s ⪇G symplectic automorphisms is among 11 maximal ones as classified by Mukai.

Let Γ be a lattice in G = SL(n,R) and X = G/S a homogeneous space of G, where S is a closed subgroup of G which contains a real algebraic subgroup H such that G/H is compact. We establish uniform distribution of orbits of Γ in X analogous to the classical equidistribution on torus. To obtain this result, we first prove an ergodic theorem along balls in the connected component of Borel subgroup ...

It has been shown that there is a Hamilton cycle in every connected Cayley graph on any group G whose commutator subgroup is cyclic of prime-power order. This note considers connected, vertex-transitive graphs X of order at least 3, such that the automorphism group of X contains a vertex-transitive subgroup G whose commutator subgroup is cyclic of prime-power order. We show that of these graphs...