A new Euclidean tight 6-design

We give a new example of Euclidean tight 6-design in $\mathbb R^{22}$.Comment: 9 page

On Turing dynamical systems and the Atiyah problem

Main theorems of the article concern the problem of M. Atiyah on possible values of l^2-Betti numbers. It is shown that all non-negative real numbers are l^2-Betti numbers, and that "many" (for example all non-negative algebraic) real numbers are l^2-Betti numbers of simply connected manifolds with respect to a free cocompact action. Also an explicit example is constructed which leads to a simply connected manifold with a transcendental l^2-Betti number with respect to an action of the threefold direct product of the lamplighter group Z/2 wr Z. The main new idea is embedding Turing machines into integral group rings. The main tool developed generalizes known techniques of spectral computations for certain random walk operators to arbitrary operators in groupoid rings of discrete measured groupoids.Comment: 35 pages; essentially identical to the published versio

Largeness and SQ-universality of cyclically presented groups

Largeness, SQ-universality, and the existence of free subgroups of rank 2 are measures of the complexity of a finitely presented group. We obtain conditions under which a cyclically presented group possesses one or more of these properties. We apply our results to a class of groups introduced by Prishchepov which contains, amongst others, the various generalizations of Fibonacci groups introduced by Campbell and Robertson

On the expressiveness of forwarding in higher-order communication

Abstract. In higher-order process calculi the values exchanged in communications may contain processes. There are only two capabilities for received processes: execution and forwarding. Here we propose a limited form of forwarding: output actions can only communicate the parallel composition of statically known closed processes and processes received through previously executed input actions. We study the expressiveness of a higher-order process calculus featuring this style of communication. Our main result shows that in this calculus termination is decidable while convergence is undecidable.

Peripheral fillings of relatively hyperbolic groups

A group theoretic version of Dehn surgery is studied. Starting with an arbitrary relatively hyperbolic group $G$ we define a peripheral filling procedure, which produces quotients of $G$ by imitating the effect of the Dehn filling of a complete finite volume hyperbolic 3--manifold $M$ on the fundamental group $\pi_1(M)$. The main result of the paper is an algebraic counterpart of Thurston's hyperbolic Dehn surgery theorem. We also show that peripheral subgroups of $G$ 'almost' have the Congruence Extension Property and the group $G$ is approximated (in an algebraic sense) by its quotients obtained by peripheral fillings. Various applications of these results are discussed.Comment: The difference with the previous version is that Proposition 3.2 is proved for quasi--geodesics instead of geodesics. This allows to simplify the exposition in the last section. To appear in Invent. Mat

Universal deformation rings for the symmetric group S_5 and one of its double covers

Let $S_5$ denote the symmetric group on 5 letters, and let $\hat{S}_5$ denote a non-trivial double cover of $S_5$ whose Sylow 2-subgroups are generalized quaternion. We determine the universal deformation rings $R(S_5,V)$ and $R(\hat{S}_5,V)$ for each mod 2 representation $V$ of $S_5$ that belongs to the principal 2-modular block of $S_5$ and whose stable endomorphism ring is given by scalars when it is inflated to $\hat{S}_5$. We show that for these $V$, a question raised by the first author and Chinburg concerning the relation of the universal deformation ring of $V$ to the Sylow 2-subgroups of $S_5$ and $\hat{S}_5$, respectively, has an affirmative answer.Comment: 10 pages, 5 figures; the proof of Theorem 1.1(a) has been shortene

On Measuring Non-Recursive Trade-Offs

We investigate the phenomenon of non-recursive trade-offs between descriptional systems in an abstract fashion. We aim at categorizing non-recursive trade-offs by bounds on their growth rate, and show how to deduce such bounds in general. We also identify criteria which, in the spirit of abstract language theory, allow us to deduce non-recursive tradeoffs from effective closure properties of language families on the one hand, and differences in the decidability status of basic decision problems on the other. We develop a qualitative classification of non-recursive trade-offs in order to obtain a better understanding of this very fundamental behaviour of descriptional systems

Well-quasi-ordering versus clique-width : new results on bigenic classes.

Daligault, Rao and Thomassé conjectured that if a hereditary class of graphs is well-quasi-ordered by the induced subgraph relation then it has bounded clique-width. Lozin, Razgon and Zamaraev recently showed that this conjecture is not true for infinitely defined classes. For finitely defined classes the conjecture is still open. It is known to hold for classes of graphs defined by a single forbidden induced subgraph H, as such graphs are well-quasi-ordered and are of bounded clique-width if and only if H is an induced subgraph of P4P4. For bigenic classes of graphs i.e. ones defined by two forbidden induced subgraphs there are several open cases in both classifications. We reduce the number of open cases for well-quasi-orderability of such classes from 12 to 9. Our results agree with the conjecture and imply that there are only two remaining cases to verify for bigenic classes

Locally finite groups containing a 2 -element with Chernikov centralizer

Suppose that a locally finite group G has a 2-element g with Chernikov centralizer. It is proved that if the involution in ⟨g⟩ has nilpotent centralizer, then G has a soluble subgroup of finite index