# A mixed identity-free elementary amenable group

@article{Jacobson2019AMI, title={A mixed identity-free elementary amenable group}, author={Brian Jacobson}, journal={Communications in Algebra}, year={2019}, volume={49}, pages={235 - 241} }

Abstract A group G is called mixed identity-free if for every and every there exists a homomorphism such that is the identity on G and is nontrivial. In this paper, we make a modification to the construction of elementary amenable lacunary hyperbolic groups provided by Ol’shanskii et al. to produce finitely generated elementary amenable groups which are mixed identity-free. As a byproduct of this construction, we also obtain locally finite p-groups which are mixed identity-free.

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## References

SHOWING 1-10 OF 14 REFERENCES

### MIXED IDENTITIES AND MIXED VARIETIES OF GROUPS

- Mathematics
- 1987

A mixed identity in variables over a group is a word (where the coefficients lie in , , and ) taking the value 1 for any values of the variables in . The concept of a mixed variety of groups is…

### Elementary amenable groups

- Mathematics
- 1980

1. In order to explain the Hausdorff-Banach-Tarski paradox, von Neumann [19] introduced the class of amenable groups in 1929. Since then the theory of amenable groups has advanced in many fronts, for…

### Transitivity degrees of countable groups and acylindrical hyperbolicity

- Mathematics
- 2015

We prove that every countable acylindrically hyperbolic group admits a highly transitive action with finite kernel. This theorem uniformly generalizes many previously known results and allows us to…

### Degrees of Growth of Finitely Generated Groups, and the Theory of Invariant Means

- Mathematics
- 1985

This paper gives a negative solution to the problem of Milnor concerning the degrees of growth of groups. The construction also answers a question of Day concerning amenable groups. A number of other…

### Lacunary hyperbolic groups

- Mathematics
- 2007

We call a finitely generated group lacunary hyperbolic if one of its asymptotic cones is an R‐tree. We characterize lacunary hyperbolic groups as direct limits of Gromov hyperbolic groups satisfying…

### ON THE GIRTH OF GROUPS

- Mathematics
- 2001

DISCLAIMER: Everything that follows is of a preliminary nature. We give a new invariant for finitely generated groups, called the girth. Several results which indicate that the girth of a group might…

### TOPICS IN GEOMETRIC GROUP THEORY

- Mathematics
- 2015

We present a brief overview of methods and results in geometric group theory, with the goal of introducing the reader to both topological and metric perspectives. Prerequisites are kept to a minimum:…

### Lacunary hyperbolic groups With an appendix by Michael Kapovich and Bruce Kleiner

- Geom. & Topol
- 2009