Automaton groups and complete square complexes

Isotopies of 3-manifolds

This paper is concerned with the homotopy theory and simple homotopy theory of connected finite 2 dimensional CW complexes with finite cyclic fundamental group. The main theorem presents a complete classification of such complexes up to homotopy type, and this theorem has the corollary that homotopy type and simple homotopy type coincide for these complexes. This work is motivated by the general problem of describing the sets HT(n) and SHT(n) of homotopy types and simple homotopy types of n-complexes, that is, connected finite 2 dimensional CW-complexes with a given fundamental group n. A visually satisfying description of HT (n) or SHT (n) is that of either set as a graph whose edges connect the type of each n-complex X to the type of its sum X v S 2 with the 2-sphere S 2. These graphs are actually trees; they clearly contain no circuits, and they are connected because any two n-complexes have the same type once each is summed with an appropriate number of copies of the 2-sphere S 2. To re establish this latter observation ofJ. H. C. Whitehead ([20, Theorem 12]), note that each n-complex has the simple homotopy type of one modeled in an obvious fashion on some finite presentation of the fundamental group ~z (see Proposition 1). But two finite presentations of the same group n differ by a finite sequence of Tietze operations, two of which leave the simple homotopy type of the associated topological model unchanged, while two alter the simple homotopy type by an S 2 summand. Of special interest in each of these trees are the roots and the junctions. The roots are the (simple) homotopy types that do not admit a factorization involving an S 2 summand; they generate the rest of the types in the tree under the operation of forming sum with S 2. The junctions are the (simple) homotopy types that admit two or more inequivalent factorizations involving an S 2 summand; they determine the shape of the tree. Each junction is a 2-dimensional instance of non-cancellation of the 2-sphere S 2 with respect to the sum operation. When the group n is a free group F of finite rank or is the finite cyclic group Z~ of prime order q, complete descriptions of the trees HT (n) and SHT (n) can be derived from the literature, as follows. A result of C. T. C. Wall ([17, Proposition 3.3]) can be specialized to read that for a free group F of finite rank r every F-complex has the homotopy type of a sum of r

The Yamabe invariant is an invariant of a closed smooth manifold defined using conformal geometry and the scalar curvature. Recently, Petean showed that the Yamabe invariant is nonnegative for all closed simply connected manifolds of dimension ≥ 5. We extend this to show that Yamabe invariant is nonnegative for all closed manifolds of dimension ≥ 5 with fundamental group of odd order having all Sylow subgroups abelian. The main new geometric input is a way of studying the Yamabe invariant on Toda brackets. A similar method of proof shows that all closed oriented manifolds of dimension ≥ 5 with non-spin universal cover, with finite fundamental group having all Sylow subgroups elementary abelian, admit metrics of positive scalar curvature, once one restricts to the complement of manifolds whose homology classes are toral. The exceptional toral homology classes only exist in dimensions not exceeding the rank of the fundamental group, so this proves important cases of the Gromov-Lawson-Rosenberg Conjecture once the dimension is sufficiently large.

The dominant topologies in the Euclidean path integral for quantum gravity differ sharply according to the sign of the cosmological constant. For , saddle points can occur only for topologies with vanishing first Betti number and finite fundamental group. For , on the other hand, the path integral is dominated by topologies with extremely complicated fundamental groups; while the contribution of each individual manifold is strongly suppressed, the `density of topologies' grows fast enough to overwhelm this suppression. The value is thus a sort of boundary between phases in the sum over topologies. I discuss some implications for the cosmological constant problem and the Hartle-Hawking wavefunction.

We present an algorithm for the following problem: given a context-free grammar for the word problem of a virtually free group G, compute a finite graph of groups G with finite vertex groups and fundamental group G. Our algorithm is non-deterministic and runs in doubly exponential time. It follows that the isomorphism problem of context-free groups can be solved in doubly exponential space. Moreover, if, instead of a grammar, a finite extension of a free group is given as input, the construction of the graph of groups is in NP and, consequently, the isomorphism problem in PSPACE.

In [3] we initiated our study of the automorphism groups of a certain class of near-rings. Specifically, let P be any complex polynomial and let P denote the near-ring of all continuous selfmaps of the complex plane where addition of functions is pointwise and the product fg of two functions f and g in P is defined by fg=f∘P∘g. The near-ring P is referred to as a laminated near-ring with laminating element P. In [3], we characterised those polynomials P(z)=anzn + an−1zn−1 +…+a0 for which Aut P is a finite group. We are able to show that Aut P is finite if and only if Deg P≧3 and ai ≠ 0 for some i ≠ 0, n. In addition, we were able to completely determine those infinite groups which occur as automorphism groups of the near-rings P. There are exactly three of them. One is GL(2) the full linear group of all real 2×2 nonsingular matrices and the other two are subgroups of GL(2). In this paper, we begin our study of the finite automorphism groups of the near-rings P. We get a result which, in contrast to the situation for the infinite automorphism groups, shows that infinitely many finite groups occur as automorphism groups of the near-rings under consideration. In addition to this and other results, we completely determine Aut P when the coefficients of P are real and Deg P = 3 or 4.

Let $X$ be a finite CW-complex with finite fundamental group. We show that the group ${\text {aut}}(X)$ of homotopy classes of self-homotopy equivalences of $X$ is commensurable to an arithmetic group. If in addition $X$ is an oriented manifold then the subgroup ${\text {au}}{{\text {t}}_t}(X)$ of homotopy classes of tangential homotopy equivalences is commensurable to an arithmetic group. Moreover if $X$ is a smooth manifold of dimension $\geqslant 5$ then the subgroup ${\text {diff}}(X)$ of ${\text {aut}}(X)$ the elements of which are represented by diffeomorphisms is also commensurable to an arithmetic group.

Coherence, subgroup separability, and metacyclic structures for a class of cyclically presented groups

Weights for symmetric and general linear groups

Let (S, º) be a finite simple group, s i , i = 1, ... , n , be fixed (not necessarily distinct) elements of S , and let E α 1 , E α 2 , ... , E α k +1 be a random realisation of a chain of states of a simple homogeneous irreducible Markov chain with the set of states { E 1 , E 2 , ... , E n }. We study convergence conditions and limit distributions for the sequences of random products of the form η ( k ) = s α 1 º s α 2 º ... º s α k +1 . The convergence conditions are formulated in terms of some homomorphism from the fundamental group of the transition graph of the Markov chain to the structural group of the semigroup S .

We give a complete classification of the Dehn surgeries on Montesinos knots which yield manifolds with cyclic or finite fundamental groups.

Linkings and Pfaff's equations D Bennequin The topology of four-dimensional manifolds Yu P Solov'ev Singularities of systems of rays Vladimir I Arnol'd On the geometry of hyperbolic attractors of smooth cascades Romen V Plykin Surgery on manifolds with finite fundamental groups A F Kharshiladze Embedding of compacta, stable homotopy groups of spheres, and singularity theory P M Akhmet'ev Cell-like maps. The problem of raising dimension A N Dranishnikov and E V Shchepin IMMERSIONS AND EMBEDDINGS OF SMOOTH MANIFOLDS

Homogeneous symplectic manifolds with Ricci-type curvature

In this paper, we develop the theory of probabilistic variants of the one-category and diagonal topological complexity, which bound the classical LS-category and topological complexity from below. Unlike any other classical or probabilistic invariants, these invariants are rigid on spaces with finite fundamental group. On Eilenberg–Mac Lane spaces, we identify these new invariants with distributional category and complexity, respectively, and use them to illuminate aspects of the behavior of the latter invariants on aspherical spaces and products of spaces. We also study their properties on covering maps, [Formula: see text]-isomorphisms, [Formula: see text]-spaces, and closed essential manifolds, and consequently, obtain the first examples of closed manifolds beyond the real projective spaces on which the distributional theory disagrees with the classical one.

We prove that any compact complex manifold with finite fundamental group and algebraic dimension zero admits no holomorphic affine connection.