Embeddable anticonformal automorphisms of Riemann surfaces

Let S be a Riemann surface and f be an automorphism of finite order of S. We call f embeddable if there is a conformal embedding such that is the restriction to e(S) of a rigid motion. In this paper we show that an anticonformal automorphism of finite order is embeddable if and only if it belongs to one of the topological conjugation classes here described. For conformal automorphisms a similar result was known by R.A. Rüedy [R3]. Received: February 8, 1996     

On the Antipode of a Co-Frobenius (Co)Quasitriangular Hopf Algebra

For H a quasitriangular Hopf algebra, S ², the square of the antipode is the inner automorphism induced by the Drinfeld element u, and S ⁴ is the inner automorphism induced by the grouplike element g = uS(u)^?1. For H finite dimensional, results of Drinfeld and Radford express g in terms of the modular elements of H. This note supplies another proof which replaces the requirement of finite dimensionality with existence of a nonzero integral for H in H*. Similar results hold for the infinite dimensional coquasitriangular case; here we supply some interesting examples.     

Twisted Conjugacy Classes for Polyfree Groups

Let G be a finitely generated polyfree group. If G has nonzero Euler characteristic then we show that Aut(G) has a finite index subgroup in which every automorphism has infinite Reidemeister number. For certain G of length 2, we show that the number of Reidemeister classes of every automorphism is infinite.     

Uniform Uniform Exponential Growth of Subgroups of the Mapping Class Group

Let Mod(S) denote the mapping class group of a compact, orientable surface S. We prove that finitely generated subgroups of Mod(S) which are not virtually abelian have uniform exponential growth with minimal growth rate bounded below by a constant depending only, and necessarily, on S. For the proof, we find in any such subgroup explicit free group generators which are “short” in any word metric. Besides bounding growth, this allows a bound on the return probability of simple random walks.     

Class-Preserving Coleman Automorphisms of Finite Groups

 Let G be a finite group whose Sylow 2-subgroups are either cyclic, dihedral, or generalized quaternion. It is shown that a class-preserving automorphism of G of order a power of 2 whose restriction to any Sylow subgroup of G equals the restriction of some inner automorphism of G is necessarily an inner automorphism. Interest in such automorphisms arose from the study of the isomorphism problem for integral group rings, see [6, 7, 13, 14].     

Automorphisms of -subshifts of finite type

Let (Σ, σ) be a ^d-subshift of finite type. Under a strong irreducibility condition (strong specification), we show that Aut(Σ) contains any finite group. For ^d-subshift of finite type without strong specification, examples show that topological mixing is not sufficient to give any finite group in the automorphism group in general: in particular, End(Σ) may be an abelian semigroup. For an example of a topologically mixing ²-subshift of finite type, the endomorphism semigroup and automorphism group are computed explicitly. This subshift has periodic-point permutations that do not extend to automorphisms.     

The Teichmüller distance between finite index subgroups of $${PSL_2(\mathbb{Z})}$$

For a given , we show that there exist two finite index subgroups of which are -quasisymmetrically conjugated and the conjugation homeomorphism is not conformal. This implies that for any there are two finite regular covers of the Modular once punctured torus T ₀ (or just the Modular torus) and a -quasiconformal map between them that is not homotopic to a conformal map. As an application of the above results, we show that the orbit of the basepoint in the Teichmüller space T(S ^p ) of the punctured solenoid S ^p under the action of the corresponding Modular group (which is the mapping class group of S ^p [6], [7]) has the closure in T(S ^p ) strictly larger than the orbit and that the closure is necessarily uncountable.      

Class-Preserving Coleman Automorphisms of Finite Groups

 Let G be a finite group whose Sylow 2-subgroups are either cyclic, dihedral, or generalized quaternion. It is shown that a class-preserving automorphism of G of order a power of 2 whose restriction to any Sylow subgroup of G equals the restriction of some inner automorphism of G is necessarily an inner automorphism. Interest in such automorphisms arose from the study of the isomorphism problem for integral group rings, see [6, 7, 13, 14]. Received 30 September 2001; in revised form 10 December 2001     

On the concept of level for subgroups of SL2 over arithmetic rings

We define the concept of level for arbitrary subgroups Γ of finite index in the special linear group SL₂(A _S), whereA _S is the ring ofS-integers of a global fieldk provided thatk is an algebraic number field, or card (S) ≥ 2. It is shown that this concept agrees with the usual notion of ‘Stufe’ for congruence subgroups. In the case SL₂(O),O the ring of integers of an imaginary quadratic number field, this criterion for deciding whether or not an arbitrary subgroup of finite index is a congruence subgroup is used to determine the minimum of the indices of non-congruence subgroups.     

The picard group of a category of graded modules

For R a G-graded ring, we study Pic(R-gr), the group of isomorphism classes of autoequivalences of the category of graded left R-modules. For G infinite, this requires generalizing the classical sequences involving Pic(A), A a fc-algebra, to A a ring with local units. Then for G either finite or infinite, we characterize the inner automorphisms in some subgroups H of the automorphism group of the smash product R#PG and thus obtain some subgroups of Pic(R-gr).     

Guts of surfaces in punctured-torus bundles

Let M be a hyperbolic manifold of finite volume which fibers over the circle with fiber a once punctured torus, and let S be an arbitrary incompressible surface in M. We determine the characteristic Jaco-Shalen-Johannson-submanifold of M−S and show, in particular, that Guts (M, S) is empty. Received: 02 April 2004     

有限ATI-群的类保持Coleman自同构

设G是一个有限群,对G的任意阿贝尔子群A及任意g∈G,若A∩A~g=1或A,则称G为一个ATI-群.本文证明了,对任意p∈τ(G),如果ATI-群G的一个p-方幂阶类保持自同构在G的任意Sylow子群上的限制等于G的某个内自同构的限制,则它必定是一个内自同构.作为该结果的一个直接推论,我们也证明了有限ATI-群G有正规化性质.     

Gale duality and homogeneous toric varieties

A non-degenerate toric variety X is called S-homogeneous if the subgroup of the automorphism group Aut(X) generated by root subgroups acts on X transitively. We prove that maximal S-homogeneous toric varieties are in bijection with pairs (P,𝒜), where P is an abelian group and 𝒜 is a finite collection of elements in P such that 𝒜 generates the group P and for every a∈𝒜 the element a is contained in the semigroup generated by 𝒜?{a}. We show that any non-degenerate homogeneous toric variety is a big open toric subset of a maximal S-homogeneous toric variety. In particular, every homogeneous toric variety is quasiprojective. We conjecture that any non-degenerate homogeneous toric variety is S-homogeneous.     

Abelian Subgroups of Garside Groups

In this article, we show that for every abelian subgroup H of a Garside group, some conjugate g ^?1 Hg consists of ultra summit elements and the centralizer of H is a finite index subgroup of the normalizer of H. Combining with the results on translation numbers in Garside groups, we obtain an easy proof of the algebraic flat torus theorem for Garside groups and solve several algorithmic problems concerning abelian subgroups of Garside groups.     

Automorphism groups of graphs with forbidden subgraphs

For a graph Ф letF(Ф) be the class of finite graphs which do not contain an induced subgraph isomorphic to Ф. We show that whenever Ф is not isomorphic to a path on at most 4 vertices or to the complement of such a graph then for every finite groupG there exists a graph ГєF(Ф) such thatG is isomorphic to the automorphism group of Г. For all paths д on at most 4 vertices we determine the class of all automorphism groups of members ofF(д).     

Unitary Representations of Oligomorphic Groups

We obtain a complete classification of the continuous unitary representations of oligomorphic permutation groups (those include the infinite permutation group S ^??, the automorphism group of the countable dense linear order, the homeomorphism group of the Cantor space, etc.). Our main result is that all irreducible representations of such groups are obtained by induction from representations of finite quotients of open subgroups and, moreover, every representation is a sum of irreducibles. As an application, we prove that many oligomorphic groups have property (T). We also show that the Gelfand?CRaikov theorem holds for topological subgroups of S ^??: for all such groups, continuous irreducible representations separate points in the group.     

The combinatorics and the homology of the poset of subgroups of p-power index

For a finite group G and a prime p the poset S^p (G) of all subgroups H ≠ G of p-power index is studied. The Möbius number of the poset is given and the homotopy type of the poset is determined as a wedge of spheres. We describe the representation of G on the homology groups of the order complex of S^p (G) and show that this representation can be realized by matrices with entries in the set {+1, -1, 0}. Finally a CL-shellable subposet of S^p (G) is exhibited for odd primes p.     

Adelic version of Margulis arithmeticity theorem

We generalize Margulis's S-arithmeticity theorem to the case when S can be taken as an infinite set of primes. Let R be the set of all primes including infinite one and set . Let S be any subset of R. For each , let be a connected semisimple adjoint -group and be a compact open subgroup for each finite prime . Let denote the restricted topological product of 's, with respect to 's. Note that if S is finite, . We show that if , any irreducible lattice in is a rational lattice. We also present a criterion on the collections and for to admit an irreducible lattice. In addition, we describe discrete subgroups of generated by lattices in a pair of opposite horospherical subgroups. Received: 30 November 2000 / Revised version: 2 April 2001 / Published online: 24 September 2001     

Solvable Subgroups of Out(F n ) are Virtually Abelian

Let F _n be the free group of rank n, let Aut(F _n ) be its automorphism group and let Out(F _n ) be its outer automorphism group. We show that every solvable subgroup of Out(F _n ) has a finite index subgroup that is finitely generated and free Abelian. We also show that every Abelian subgroup of Out(F _n ) has a finite index subgroup that lifts to Aut(F _n ).