Group equivariant operators are playing a more and relevant role in machine learning topological data analysis. In this paper, we present some new results concerning the construction of G-equivariant non-expansive (GENEOs) from space \(\Phi \) real-valued bounded continuous functions on X to itself. The represents our set data, while G is subgroup group all self-homeomorphisms X, representing i...

1. Reminder on quotient groups Let G be a group and H a subgroup of G. We have seen that the left cosets xH of H in G give a partition of G. Motivated by the case of quotients of rings by ideals, it is natural to consider the product operation on cosets. Recall that for any subsets S, T of G, by ST we mean {st | s ∈ S, t ∈ T }. If G is commutative, the product of two left cosets is another left...

Let X be a compact Kähler manifold of complex dimension n, and let G be a connected solvable subgroup of the automorphism group Aut(X). Let N(G) be the normal subgroup of G of elements of null entropy. The aim of this short paper is to show that G/N(G) is a free abelian group of rank ≤ n− 1 and that the rank estimate is optimal. This proves the conjecture of Tits type in [6] affirmatively. The ...

‎we characterize those groups $g$ and vector spaces $v$ such that $v$ is a faithful irreducible $g$-module and such that each $v$ in $v$ is centralized by a $g$-conjugate of a fixed non-identity element of the fitting subgroup $f(g)$ of $g$‎. ‎we also determine those $v$ and $g$ for which $v$ is a faithful quasi-primitive $g$-module and $f(g)$ has no regular orbit‎. ‎we do use these to show in ...

with K, n0 ≥ 1, (X n )n≥0 identically distributed as (Xn)n≥0 for r = 1, . . . , K , a random vector I (n) = (I (n) 1 , . . . , I (n) K ) of integers in {0, . . . , n − 1} and a random bn such that (X n )n≥0, . . . , (X ) n )n≥0, (I (n), bn) are independent. The symbol d = denotes equality in distribution. In applications, the I (n) r are random subgroup sizes, bn is a toll function specifying t...

We give a short and uniform proof of a special case of Tits’ Centre Conjecture using a theorem of J-P. Serre [8] and a result from [1]. Precisely, we show that this conjecture holds for fixed point subcomplexes X of the building X = X(G) of a connected reductive algebraic group G, where H is a subgroup of G.

We use cross ratios to describe second real continuous bounded cohomology for locally compact σ-compact topological groups. We also investigate the second continuous bounded cohomology group of a closed subgroup of the isometry group Iso(X) of a proper hyperbolic geodesic metric space X and derive some rigidity results for Iso(X)-valued cocycles.

We show that a group G acts properly and effectively on a locally compact and σ-compact metric space (X, d) if and only if there exists a compatible G-invariant Heine-Borel metric dp on X such that G is homeomorphic to a closed subgroup of the group of isometries Iso(X, dp).

We prove the following version of W lodarczyk’s Embedding Theorem: Every normal complex algebraic C∗-variety Y admits an equivariant closed embedding into a toric prevariety X on which C∗ acts as a one-parameter-subgroup of the big torus T ⊂ X. If Y is Q-factorial, then X may be chosen to be simplicial and of affine intersection.