Given the large class of groups already known to be sofic, there is seemingly a shortfall in results concerning their permanence properties. We address this problem for wreath products, and in particular investigate the behaviour of more general metric approximations of groups under wreath products.Our main result is the following. Suppose that H is a sofic group and G is a countable, discrete group. If G is sofic, hyperlinear, weakly sofic, or linear sofic, then G H is also sofic, hyperlinear, weakly sofic, or linear sofic respectively. In each case we construct relevant metric approximations, extending a general construction of metric approximations for G H that uses soficity of H.

37 pages, 4 figuresInternational audienceOuter automorphism groups of RAAGs, denoted $Out(A_\Gamma)$, interpolate between $Out(F_n)$ and $GL_n(\mathbb{Z})$. We consider several vastness properties for which $Out(F_n)$ behaves very differently from $GL_n(\mathbb{Z})$: virtually mapping onto all finite groups, SQ-universality, virtually having an infinite dimensional space of homogeneous quasimorphisms, and not being boundedly generated. We give a neccessary and sufficient condition in terms of the defining graph $\Gamma$ for each of these properties to hold. Notably, the condition for all four properties is the same, meaning $Out(A_\Gamma)$ will either satisfy all four, or none. In proving this result, we describe conditions on $\Gamma$ that imply $Out(A_\Gamma)$ is large. Techniques used in this work are then applied to the case of McCool groups, defined as subgroups of $Out(F_n)$ that preserve a given family of conjugacy classes. In particular we show that any McCool group that is not virtually abelian virtually maps onto all finite groups, is SQ-universal, is not boundedly generated, and has a finite index subgroup whose space of homogeneous quasimorphisms is infinite dimensional

Abstract. In this paper, we describe an effective version of the conjugacy problem and study it for wreath products and free solvable groups. The problem involves estimating the length of short conjugators between two elements of the group, a notion which leads to the definition of the conjugacy length function. We show that for free solvable groups the conjugacy length function is at most cubic. For wreath products the behaviour depends on the conjugacy length function of the two groups involved, as well as subgroup distortion within the quotient group.In geometric group theory there has often been a tendency to produce more effective results. For example, calculating the Dehn function of a group is an effective version of the word problem and it gives us a better understanding of its complexity. We can, furthermore, use this extra information to determine more details of the group at hand. Estimating the length of short conjugators in a group could be described as an effective version of the conjugacy problem, and finding a control on these lengths in wreath products and free solvable group is the main motivation of this paper.The conjugacy problem is one of Max Dehn's three decision problems for groups formulated in 1912 (see [5]). Dehn originally described these problems because of the significance he discovered they had in the geometry of 3-manifolds and they have since become the most fundamental problems in combinatorial and geometric group theory. Let be a finitely presented group with finite generating set X . The conjugacy problem asks whether there is an algorithm which determines when two given words on X [ X 1 represent conjugate elements in . This question may also be asked of recursively presented groups, and we can try to develop our understanding further by asking whether one can find, in some sense, a short conjugator between two given conjugate elements of a group.Suppose word-lengths in , with respect to the given generating set X, are denoted by j j. The conjugacy length function is the minimal function CLF W N ! N

Abstract. A group has finite palindromic width if there exists n such that every element can be expressed as a product of n or fewer palindromic words. We show that if G has finite palindromic width with respect to some generating set, then so does G ≀ Z r . We also give a new, self-contained, proof that finitely generated metabelian groups have finite palindromic width. Finally, we show that solvable groups satisfying the maximal condition on normal subgroups (max-n) have finite palindromic width.

No abstract