Solvability of the conjugacy problem for finite subgroups in automorphism groups of free groups

We prove that there exists an algorithm which solves a conjugacy problem for finite subgroups in automorphism and outer automorphism groups of a free group of finite rank. Of independent interest is the construction of an algorithm of decomposing an arbitrary free-by-finite group into a fundamental group of a finite graph of finite groups, with the number of steps evaluated explicitly. In passing, we solve the conjugacy problem for finite subgroups in almost free groups. As a consequence, an algorithm is obtained computing generating sets for a group of fixed points in an arbitrary finite automorphism group of a free group of finite rank.Translated fromAlgebra i Logika, Vol. 34, No. 5, pp. 558–606, September-October, 1995.Supported by the RFFR grant No. 93-011-1508 and by the ISF (International Science Foundation) grant RPC000.     

Two-Generator Arithmetic Kleinian Groups II

Extending earlier work, we establish the finiteness of the numberof two-generator arithmetic Kleinian groups with one generatorparabolic and the other either parabolic or elliptic. We alsoidentify all the arithmetic Kleinian groups generated by twoparabolic elements. Surprisingly, there are exactly 4 of these,up to conjugacy, and they are all torsion free. 1991 MathematicsSubject Classification 30F40, 20H10.     

The conjugacy problem for automorphism groups of countable homogeneous structures

We consider the conjugacy problem for the automorphism groups of a number of countable homogeneous structures. In each case we find the precise complexity of the conjugacy relation in the sense of Borel reducibility.     

The conjugacy problem for Dehn twist automorphisms of free groups

A Dehn twist automorphism of a group G is an automorphism which can be given (as specified below) in terms of a graph-of-groups decomposition of G with infinite cyclic edge groups. The classic example is that of an automorphism of the fundamental group of a surface which is induced by a Dehn twist homeomorphism of the surface. For , a non-abelian free group of finite rank n, a normal form for Dehn twist is developed, and it is shown that this can be used to solve the conjugacy problem for Dehn twist automorphisms of . Received: February 12, 1996.     

Nonabelian 1-cohomologies and conjugacy of finite subgroups in some extensions

We describe the conjugacy classes of finite subgroups in some split extensions using the notion of 1-cocycle and 1-coboundary with values in a noncommutative group. We prove that each finite subgroup in the automorphism group of a free Lie algebra of rank 3 is conjugated with a subgroup of the linear automorphism group provided that the group order does not divide the characteristic of the ground field.     

The conjugacy problem for the automorphism group of the random graph

We prove that the conjugacy problem for the automorphism group of the random graph is Borel complete, and discuss the analogous problem for some other countably categorical structures.     

Coding Subset Shift by Subgroup Conjugacy

We present a Borel reduction from a subset shift equivalencerelation of a countable group to a subgroup conjugacy relationof a free product. The technique gives a much shorter proofof an earlier result of Thomas and Velickovic. 1991 MathematicsSubject Classification 04A15, 20A10.     

Conjugacy separability of 1-acylindrical graphs of free groups

We use the theory of group actions on profinite trees to prove that the fundamental group of a finite, 1-acylindrical graph of free groups with finitely generated edge groups is conjugacy separable. This has several applications: we prove that positive, C′(1/6) one-relator groups are conjugacy separable; we provide a conjugacy separable version of the Rips construction; we use this latter to provide an example of two finitely presented, residually finite groups that have isomorphic profinite completions, such that one is conjugacy separable and the other does not even have solvable conjugacy problem.     

On the Bergman Property for the Automorphism Groups of Relatively Free Groups

A group G is said to have the Bergman property (the propertyof uniformity of finite width) if given any generating X withX = X^–1 of G, we have that G = X^k for some natural k,that is, every element of G is a product of at most k elementsof X. We prove that the automorphism group Aut(N) of any infinitelygenerated free nilpotent group N has the Bergman property. Also,we obtain a partial answer to a question posed by Bergman byestablishing that the automorphism group of a free group ofcountably infinite rank is a group of uniformly finite width.     

Characterizing Automorphism Groups of Ordered Abelian Groups

A proof is given of the following theorem, which characterizesfull automorphism groups of ordered abelian groups: a groupH is the automorphism group of some ordered abelian group ifand only if H is right-orderable. 2000 Mathematics Subject Classification20K15, 20K20, 20F60, 20K30 (primary); 03E05 (secondary).     

Polynomial Dehn Functions and the Length of Asynchronously Automatic Structures

We extend the range of observed behaviour among length functionsof optimal asynchronously automatic structures. We do so bymeans of a construction that yields asynchronously automaticgroups with finite aspherical presentations where the Dehn functionof the group is polynomial of arbitrary degree. Many of thesegroups can be embedded in the automorphism group of a free group.Moreover, the fact that the groups have aspherical presentationsmakes them useful tools in the search to determine the spectrumof exponents for second order Dehn functions. We contributeto this search by giving the first exact calculations of groupswith quadratic and superquadratic exponents. 2000 Mathematical Subject Classification: 20F06, 20F65, 20F69.     

Counting Growth Types of Automorphisms of Free Groups

Given an automorphism of a free group Fn, we consider the following invariants: e is the number of exponential strata (an upper bound for the number of different exponential growth rates of conjugacy classes); d is the maximal degree of polynomial growth of conjugacy classes; R is the rank of the fixed subgroup. We determine precisely which triples (e, d, R) may be realized by an automorphism of Fn. In particular, the inequality e ￡ \frac3n-24{{e \leq \frac{3n-2}{4}}} (due to Levitt–Lustig) always holds. In an appendix, we show that any conjugacy class grows like a polynomial times an exponential under iteration of the automorphism.     

The first homology of the group of equivariant diffeomorphisms and its applications

Let V be a representation space of a finite group G. We determinethe group structure of the first homology of the equivariantdiffeomorphism group of V. Then we can apply it to the calculationof the first homology of the corresponding automorphism groupsof smooth orbifolds, compact Hausdorff foliations, codimensionone or two compact foliations and the locally free S¹-actionson 3-manifolds. Received November 5, 2007.     

Twisted conjugacy classes of the unit element

We study twisted conjugacy classes of the unit element in different groups. Fel’shtyn and Troitsky showed that the twisted conjugacy class of the unit element of an abelian group is a subgroup for every automorphism. The structure is investigated of a group whose twisted conjugacy class of the unit element is a subgroup for every automorphism (inner automorphism).     

Finite extensions of free pro-P groups of rank at most two

For a pro-p groupG, containing a free pro-p open normal subgroup of rank at most 2, a characterization as the fundamental group of a connected graph of cyclic groups of order at mostp, and an explicit list of all such groups with trivial center are given. It is shown that any automorphism of a free pro-p group of rank 2 of coprime finite order is induced by an automorphism of the Frattini factor groupF/F ^* . Finally, a complete list of automorphisms of finite order, up to conjugacy in Aut(F), is given. Supported by an NSERC grant. Supported by the Austrian Science Foundation.     

Twisted conjugacy in braid groups

In this note we solve the twisted conjugacy problem for braid groups, i.e., we propose an algorithm which, given two braids u, υ ∈ B _n and an automorphism φ ∈ Aut(B _n ), decides whether υ = (φ(x))^?1 ux for some x ∈ B _n . As a corollary, we deduce that each group of the form B _n ? H, a semidirect product of the braid group B _n by a torsion-free hyperbolic group H, has solvable conjugacy problem.     

The automorphism group of a graph product of groups

A theorem of Karrass, Pietrowski and Solitar on the structure of the automorphism group of an amalgamated free product is extended to automorphism groups of fundamental groups of graphs of groups in which the edge groups are incomparable up to conjugacy.     

Exotic Actions on Trees

We give an explicit example of an exotic (non-simplicial), geometricfree action of the free group F₃ on an R-tree T. We begin byassociating an interval translation mapping of the unit intervalto an automorphism of F₃. We use a result of D. Gaboriau andG. Levitt to obtain an F₃-action on an R-tree T. We show thatfor our particular choice of , the resulting F₃-action is minimal,free and exotic. 1991 Mathematics Subject Classification 20E36.