Abstract. In this note, we announce the first results on quasi-isometric rigidity of non-nilpotent polycyclic groups. In particular, we prove that any group quasiisometric to the three dimenionsional solvable Lie group Sol is virtually a lattice in Sol. We prove analogous results for groups quasi-isometric to R⋉R n where the semidirect product is defined by a diagonalizable matrix of determinant one with no eigenvalues on the unit circle. Our approach to these problems is to first classify all self quasi-isometries of the solvable Lie group. Our classification of self quasiisometries for R⋉R n proves a conjecture made by Farb and Mosher in [FM3]. Our techniques for studying quasi-isometries extend to some other classes of groups and spaces. In particular, we characterize groups quasi-isometric to any lamplighter group, answering a question of de la Harpe [dlH]. Also, we prove that certain Diestel-Leader graphs are not quasi-isometric to any finitely generated group, verifying a conjecture of Diestel and Leader from [DL] and answering a question of Woess from [SW, Wo1]. We also prove that certain non-unimodular, nonhyperbolic solvable Lie groups are not quasi-isometric to finitely generated groups. The results in this paper are contributions to Gromov’s program for classifying finitely generated groups up to quasi-isometry [Gr2]. We introduce a new technique for studying quasi-isometries, which we refer to as coarse differentiation.

Abstract. The dead-end depth of an element g of a group with finite generating set A is the distance from g to the complement of the radius dA(1, g) closed ball, in the word metric dA. We exhibit a finitely presented group K with two finite generating sets A and B such that dead-end depth is unbounded on K with respect to A but is bounded above by two with respect to B. 1.

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Abstract. Let X be a locally finite, connected graph without vertices of degree 1. Non-backtracking random walk moves at each step with equal probability to one of the “forward ” neighbours of the actual state, i.e., it does not go back along the preceding edge to the preceding state. This is not a Markov chain, but can be turned into a Markov chain whose state space is the set of oriented edges of X. Thus we obtain for infinite X that the n-step non-backtracking transition probabilities tend to zero, and we can also compute their limit when X is finite. This provides a short proof of an old result concerning cogrowth of groups, and makes the extension of that result to arbitrary regular graphs rigorous. Even when X is non-regular, but small cycles are dense in X, we show that the graph X is non-amenable if and only if the non-backtracking n-step transition probabilities decay exponentially fast. This is a partial generalization of the cogrowth criterion for regular graphs which comprises the original cogrowth criterion for finitely generated groups of Grigorchuk and Cohen. 1

This paper continues the work announced in [EFW1] and begun in [EFW2]. For a more detailed introduction, we refer the reader to those papers. As discussed in those papers, all our theorems stated above are proved using a new technique, which we call coarse differentiation. Even though quasi-isometries have no local structure

We introduce concepts of intermediate rank for countable groups that “interpolate” between consecutive values of the classical (integer-valued) rank. Various classes of groups are proved to have intermediate rank behaviors. We are especially interested in interpolation between rank 1 and rank 2. For instance, we construct groups “of rank 7 ”. Our setting is essentially that of 4 non positively curved spaces, where concepts of intermediate rank include polynomial rank, local rank, and mesoscopic rank. The resulting framework has interesting connections to operator algebras. We prove property RD in many cases where intermediate rank occurs. This gives a new family of groups satisfying the Baum-Connes conjecture. We prove that the reduced C ∗-algebras of groups of rank 7 have stable rank 1. 4 The paper is organized along the following thematic lines. A) Rank interpolation from the viewpoint of property RD; B) Triangle polyhedra and the classical rank; C) Polynomial and exponential rank, growth rank and property RD; D) Local rank, rank 7 4, existence and classification results; E) Triangle polyhedra and property RD; F) Applications to the Baum-Connes conjecture; G) C∗-algebraic rank, stable rank, real rank; H) Mesoscopic rank. Mixed local rank.

Abstract. Let Fm be a free group with m generators and let R be its normal subgroup such that Fm/R projects onto Z. We give a lower bound for the growth rate of the group Fm/R ′ (where R ′ is the derived subgroup of R) in terms of the length ρ = ρ(R) of the shortest nontrivial relation in R. It follows that the growth rate of Fm/R ′ approaches 2m − 1 as ρ approaches infinity. This implies that the growth rate of an m-generated amenable group can be arbitrarily close to the maximum value 2m − 1. This answers an open question by P. de la Harpe. In fact we prove that such groups can be found already in the class of abelian-by-nilpotent groups as well as in the class of finite extensions of metabelian groups. 1.

Giordano, Putnam and Skau showed that topological full groups of Cantor minimal systems are complete invariants for flip conjugacy. We will completely determine the structure of normal subgroups of the topological full group. Moreover, a necessary and sufficient condition for the topological full group to be finitely generated will be given. 1

differentiation of quasi-isometries I: spaces

finitely-presented solvable group with a