# Limit groups, positive-genus towers and measure-equivalence

@article{Bridson2007LimitGP, title={Limit groups, positive-genus towers and measure-equivalence}, author={Martin R. Bridson and Michael Tweedale and Henry Wilton}, journal={Ergodic Theory and Dynamical Systems}, year={2007}, volume={27}, pages={703 - 712} }

By definition, an $\omega$-residually free tower is positive-genus if all surfaces used in its construction are of positive-genus. We prove that every limit group is virtually a subgroup of a positive-genus, $\omega$-residually free tower. By combining this construction with results of Gaboriau, we prove that elementarily free groups are measure-equivalent to free groups.

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## References

SHOWING 1-10 OF 23 REFERENCES

### Examples of groups that are measure equivalent to the free group

- MathematicsErgodic Theory and Dynamical Systems
- 2005

Measure equivalence (ME) is the measure theoretic counterpart of quasi-isometry. This field has grown considerably over the past few years, developing tools to distinguish between different ME…

### Gromov’s measure equivalence and rigidity of higher rank lattices

- Mathematics
- 1999

In this paper the notion of Measure Equivalence (ME) of countable groups is studied. ME was introduced by Gromov as a measure-theoretic analog of quasi-isometries. All lattices in the same locally…

### Limit groups as limits of free groups: compactifying the set of free groups.

- Mathematics
- 2004

We give a topological framework for the study of Sela's limit groups:limit groups are limits of free groups in a compact space of markedgroups.Many results get a natural interpretation in this…

### Subgroups of Finite Index in Free Groups

- MathematicsCanadian Journal of Mathematics
- 1949

This paper has as its chief aim the establishment of two formulae associated with subgroups of finite index in free groups. The first of these (Theorem 3.1) gives an expression for the total length…

### On Orbit Equivalence of Measure Preserving Actions

- Mathematics
- 2002

We give a brief survey of some classification results on orbit equivalence of probability measure preserving actions of countable groups. The notion of l2 Betti numbers for groups is gently…

### Diophantine geometry over groups VI: the elementary theory of a free group

- Mathematics
- 2006

Abstract.This paper is the sixth in a sequence on the structure of sets of solutions to systems of equations in a free group, projections of such sets, and the structure of elementary sets defined…

### Some new rigidity results for stable orbit equivalence

- MathematicsErgodic Theory and Dynamical Systems
- 1995

Abstract Broadly speaking, we prove that an action of a group with very little commutativity cannot be stably orbit equivalent to an action of a group with enough commutativity, assuming both actions…

### Ergodic Theory and Semisimple Groups

- Mathematics
- 1984

1. Introduction.- 2. Moore's Ergodicity Theorem.- 3. Algebraic Groups and Measure Theory.- 4. Amenability.- 5. Rigidity.- 6. Margulis' Arithmeticity Theorems.- 7. Kazhdan's Property (T).- 8. Normal…

### Diophanting geometry over groups II: Completions, closures and formal solutions

- Mathematics
- 2003

This paper is the second in a series on the structure of sets of solutions to systems of equations in a free group, projections of such sets, and the structure of elementary sets defined over a free…

### Irreducible Affine Varieties over a Free Group: II. Systems in Triangular Quasi-quadratic Form and Description of Residually Free Groups

- Mathematics
- 1998

Abstract We shall prove the conjecture of Myasnikov and Remeslennikov [ 4 ] which states that a finitely generated group is fully residually free (every finite set of nontrivial elements has…