Let G be any locally compact metrizable group. The main result of this paper, roughly stated, is that if F < G is any finitely generated free group and Γ < G any lattice, then up to a small perturbation and passing to a finite index subgroup, F is a subgroup of Γ. If G/Γ is noncompact then we require additional hypotheses that include G = SO(n, 1). MSC: 20E07, 20F65, 20F67, 22D40, 20E05.

Abel Stolz and Andreas Thom [Proc. Lond. Math. Soc. 108 (2014), pp. 73–102] stated that the lattice of normal subgroups an ultraproduct finite simple groups is always linearly ordered. This false in this form most cases for classical Lie type. We correct statement case point out a version \enquote*relative bounded generation quasisimple its implications on structure such groups.

Measure rigidity for the action of maximal horospherical subgroups on homogeneous spaces over a field of positive characteristic is proved. In the case when the lattice is uniform we prove the action of any horospherical subgroup is uniquely ergodic.

Let {gt} be a nonquasiunipotent one-parameter subgroup of a connected semisimple Lie group G without compact factors; we prove that the set of points in a homogeneous spaceG/Γ (Γ an irreducible lattice inG) with bounded {gt}-trajectories has full Hausdorff dimension. Using this we give necessary and sufficient conditions for this property to hold for any Lie group G and any lattice Γ in G.

We prove that whenever a Kac-Moody group over a finite field is a lattice of its buildings, it has a fundamental domain with respect to which the induction cocycle is L for any p∈ [1; +∞). The proof uses elementary counting arguments for root group actions on buildings. The applications are the possibility to apply some lattice superrigidity, and the normal subgroup property for Kac-Moody latti...

We analyze the problem of uniqueness characterization groups by their weak congruence lattices. discuss possibility that same algebraic lattice L acts as a group in more than one way, so corresponding diagonals are represented different elements L. If this is impossible, is, if can be interpreted single we say sharp lattice. prove many classes have In particular, connections among isomorphisms ...

We study a three dimensional Z(3)-symmetric effective theory of high temperature QCD. The exact lattice-continuum relations, needed in order to perform lattice simulations with physical parameters, are computed to order O(a0) in lattice perturbation theory. Lattice simulations are performed to determine the phase structure of a subset of the parameter space. [email protected]

We study the lattice of finite-index extensions of a given finitely generated subgroup H of a free group F . This lattice is finite and we give a combinatorial characterization of its greatest element, which is the commensurator of H . This characterization leads to a fast algorithm to compute the commensurator, which is based on a standard algorithm from automata theory. We also give a sub-exp...

quasinormal subgroups have been studied for nearly 80 years. in finite groups, questions concerning them invariably reduce to p-groups, and here they have the added interest of being invariant under projectivities, unlike normal subgroups. however, it has been shown recently that certain groups, constructed by berger and gross in 1982, of an important universal nature with regard to the existen...