Francesco G. Russo

A Note on the Schur Multiplier of a Nilpotent Lie Algebra

For a nilpotent Lie algebra L of dimension n and dim (L 2) = m ≥ 1, we find the upper bound , where M(L) denotes the Schur multiplier of L. In case m = 1, the equality holds if and only if L ≅ H(1) ⊕ A, where A is an abelian Lie algebra of dimension n − 3 and H(1) is the Heisenberg algebra of dimension 3.

People counting by learning their appearance in a multi-view camera environment

We present a people counting system that, based on the information gathered by multiple cameras, is able to tackle occlusions and lack of visibility that are typical in crowded and cluttered scenes. In our method, evidence of the foreground likelihood in each available view is obtained through a bio-inspired mechanism of self-organizing background subtraction, that is robust against well known foreground detection challenges and is able to detect both moving and stationary foreground objects. This information is gathered into a synergistic framework, that exploits the homography associated to each scene view and the scene ground plane, thus allowing to reconstruct people feet positions in a single feet map image. Finally, people counting is obtained by a k-NN classification, based on learning the count estimates from the feet maps, supported by a tracking mechanism that keeps track of people movements and of their identities along time, also enabling tolerance to occasional misdetections. Experimental results with detailed qualitative and quantitative analysis and comparisons with state-of-the-art methods are provided on publicly available benchmark datasets with different crowd densities and environmental conditions.

Some criteria for detecting capable Lie algebras

In this paper, we classify all capable nilpotent Lie algebras with derived subalgebra of dimension at most 1.

A restriction on the Schur multiplier of nilpotent Lie algebras

An improvement of a bound of Yankosky (2003) is presented in this paper, thanks to a restriction which has been recently obtained by the authors on the Schur multiplier M(L) of a finite dimensional nilpotent Lie algebra L. It is also described the structure of all nilpotent Lie algebras such that the bound is attained. An important role is played by the presence of a derived subalgebra of maximal dimension. This allows precision on the size of M(L). Among other results, applications to the non-abelian tensor square L ⊗ L are illustrated.

The probability that

The FC-center of a group G is the characteristic subgroup F of all elements whose conjugacy class is finite. If G = F, then G is called an FC-group. We show that a compact group G is an FC-group if and only if its center Z(G) is open (that is, G is center by finite) if and only if its commutator subgroup is finite (that is, G is finite by commutative). Now let G be a compact group and let p denote the Haar measure of the set of all pairs (x;y) in GG for which (x;y) = 1; this is the probability that two randomly picked elements commute. We prove that p > 0 if and only if the FC-center F of G is open and so has finite index. If these conditions are satisfied, then Z(F) is a characteristic normal abelian open subgroup of G and G is abelian by finite. For a fixed pair of positive natural numbers m and n we formulate mild conditions on the profinite component factor group G=G0 depending on m and n, and we show, assuming these conditions to be satisfied, that the conditions above are also equivalent to the proposition that the probability that the powers x m and y n commute is positive for two randomly picked elements x;y2 G. As a very simple observation we note in passing that the probability of x m and y to commute for random x;y2 G is 1 if and only if x m is central for all x 2 G. References to the history of the discussion are given at the end of the paper. MSC 2010: Primary 20C05, 20P05; Secondary 43A05.

x

Recently, we have introduced a group invariant, which is re- lated to the number of elements x and y of a nite group G such that x ^ y = 1 G^ G in the exterior square G ^ G of G. This number gives re- strictions on the Schur multiplier of G and, consequently, large classes of groups can be described. In the present paper we generalize the previous investigations on the topic, focusing on the number of elements of the form h m ^ k of H ^ K such that h m ^ k = 1 H^ K , where m 1 and H and K are arbitrary subgroups of G.

and

A p-group G of order pn (p prime, n ≥ 1) satisfies a classic Greens bound logp |M(G)| ≤ ½n(n - 1) on the order of the Schur multiplier M(G) of G. Ellis and Wiegold sharpened this restriction, proving that logp |M(G)| ≤ ½(d - 1)(n + m), where |G′| = pm(m ≥ 1) and d is the minimal number of generators of G. The first author has recently shown that logp |M(G)| ≤ ½(n + m - 2)(n - m - 1) + 1, improving not only Greens bound, but several other inequalities on |M(G)| in literature. Our main results deal with estimations with respect to the bound of Ellis and Wiegold.

y

The line of investigation of the present paper goes back to a classical work of W. H. Gustafson of the 1973, in which it is described the probability that two randomly chosen group elements commute. In the same work, he gave some bounds for this kind of probability, providing information on the group structure. We have recently obtained some generalizations of his results for finite groups. Here we improve them in the context of the compact groups.

commute in a compact group,

Several authors investigated the properties which are invariant under the passage from a group to its nonabelian tensor square. In the present note we study this problem from the viewpoint of the classes of groups and the methods allow us to prove a result of invariance for some geometric properties of discrete groups.In the nonabelian tensor product G⊗H of two groups G and H many properties pass from G and H to G⊗H. There is a wide literature for different properties involved in this passage. We look at weak conditions for which such a passage may happen. 1. Terminology and statement of the result Let G and H be two groups acting upon each other in a compatible way: (1.1) ghg′ = (( g h)), hgh′ = (( h h)), for g, g ∈ G and h, h ∈ H , and acting upon themselves by conjugation. The nonabelian tensor product G ⊗ H of G and H is the group generated by the symbols g ⊗ h with defining relations (1.2) gg ⊗ h = ( g ⊗ h)(g ⊗ h), g ⊗ hh = (g ⊗ h)( g ⊗ h). When G = H and all actions are by conjugations, G ⊗ G is called nonabelian tensor square of G. These notions were introduced in [3, 4] and some significant contributions can be found in [1, 2, 5, 6, 8, 9, 10, 12, 13]. From the defining relations in G⊗H , (1.3) κ : g⊗h ∈ G⊗H 7→ κ(g⊗h) = [g, h] ∈ [G,H ] = 〈ghgh | g ∈ G, h ∈ H〉 is an epimorphism of groups. Still from [3, 4], if G and H act trivially upon each other, then G⊗H is isomorphic to the usual tensor product G⊗ZH . If they act compatibly upon each other, then their actions induce an action of the free product G ∗H on G⊗H given by (g ⊗ h) = g ⊗ h, where x ∈ G ∗H . The exterior product G ∧H is the group obtained with the additional relation g ⊗ h = 1⊗ on G⊗H , that is, (1.4) G ∧H = (G⊗H)/D, where D = 〈g ⊗ g : g ∈ G ∩H〉. Now it is easy to check that (1.5) κ : g ∧ h ∈ G ∧H 7→ κ(g ∧ h) = [g, h] ∈ [G,H ] is a well–defined epimorphism of groups. For convenience of the reader, we recall that there is a famous commutative diagram with exact rows and central extensions as columns in [3, (1)]: It correlates the second homology group H2(G) of G with the third homology group H3(G) of G, the Whitehead’s quadratic functor Γ, the Whitehead’s function ψ and kerκ = J2(G) (see also [3, 4, 14]). Date: June 20, 2012.

COMMUTING POWERS AND EXTERIOR DEGREE OF FINITE GROUPS

The (weak) geometric simple connectivity and the quasi-simple filtration are topological notions of manifolds, which may be defined for discrete groups too. It turns out that they are equivalent for finitely presented groups, but the main problem is the absence of examples of groups which do not satisfy them. In this note we study some algebraic classes of groups with respect to these properties.