Linear groups - Malcev's theorem and Selberg's lemma
An account of two fundamental facts concerning finitely generated linear groups: Malcev's theorem on residual finiteness, and Selberg's lemma on virtual torsion-freeness.
math.GR↗
arXiv subjects
Bogdan Nica
Publications and source records attributed to Bogdan Nica.
Proper isometric actions of hyperbolic groups on $L^p$-spaces
We show that every non-elementary hyperbolic group $\G$ admits a proper affine isometric action on $L^p(\bd\G\times \bd\G)$, where $\bd\G$ denotes the boundary of $\G$ and $p$ is large enough. Our construction involves a $\G$-invariant measure on $\bd\G\times \bd\G$ analogous to the Bowen - Margulis measure from the CAT$(-1)$ setting, as well as a geometric cocycle à la Busemann. We also deduce that $\G$ admits a proper affine isometric action on the first $\ell^p$-cohomology group $H^1_{(p)}(\G)$ for large enough $p$.
math.GR↗
Finitely summable Fredholm modules for boundary actions of hyperbolic groups
We construct a family of odd, finitely summable Fredholm modules over the crossed product C*-algebra $C(\bd \G)\rtimes \G$ associated to the action of a non-elementary hyperbolic group $\G$ on its Gromov boundary $\bd \G$. These Fredholm modules all represent the same, distinguished class in K-homology, namely that of the `boundary extension' of $C(\bd \G)\rtimes \G$ associated to the Gromov compactification of $\G$, and is typically nonzero. Their summability is closely related to the Hausdorff dimension of the boundary. We use these results to compute the Connes-Chern character of the boundary extension in cyclic cohomology.
math.OA↗
Relatively spectral morphisms and applications to K-theory
Spectral morphisms between Banach algebras are useful for comparing their K-theory and their "noncommutative dimensions" as expressed by various notions of stable ranks. In practice, one often encounters situations where the spectral information is only known over a dense subalgebra. We investigate such relatively spectral morphisms. We prove a relative version of the Density Theorem regarding isomorphism in K-theory. We also solve Swan's problem for the connected stable rank, in fact for an entire hierarchy of higher connected stable ranks that we introduce.
math.OA↗
Homotopical stable ranks for Banach algebras
The connected stable rank and the general stable rank are homotopy invariants for Banach algebras, whereas the Bass stable rank and the topological stable rank should be thought of as dimensional invariants. This paper studies the two homotopical stable ranks, viz. their general properties as well as specific examples and computations. The picture that emerges is that of a strong affinity between the homotopical stable ranks, and a marked contrast with the dimensional ones.
math.OA↗
On the degree of rapid decay
A finitely generated group $\G$ equipped with a word-length is said to satisfy property RD if there are $C, s\geq 0$ such that, for all non-negative integers $n$, we have $\|a\|\leq C (1+n)^s \|a\|_2$ whenever $a\in\C\G$ is supported on elements of length at most $n$. We show that, for infinite $\G$, the degree $s$ is at least 1/2.
math.GR↗
Spectral morphisms, K-theory, and stable ranks
We give a brief account of the interplay between spectral morphisms, K-theory, and stable ranks in the context of Banach algebras.
math.OA↗
Group actions on median spaces
We investigate the geometry of median metric spaces. The group-theoretic applications are towards Kazhdan's property (T) and Haagerup's property.
math.GR↗
Cubulating spaces with walls
We describe a correspondence between spaces with walls and CAT(0) cube complexes.
math.GT↗