• Corpus ID: 249926554

Subsets of Virtually Nilpotent Groups with the SBM Property

@inproceedings{Burkhart2022SubsetsOV,
  title={Subsets of Virtually Nilpotent Groups with the SBM Property},
  author={Ryan Burkhart and Isaac Goldbring},
  year={2022}
}
. We extend Leth’s notion of subsets of the integers satisfying the Standard interval measure (SIM) property to the class of virtually nilpotent groups and name the corresponding property the Standard ball measure (SBM) property. In order to do this, we define a natural measure on closed balls in asymptotic cones associated to such groups and show that this measure satisfies the Lebesgue density theorem. We then prove analogs of various properties known to hold for SIM sets in this broader… 

References

SHOWING 1-10 OF 21 REFERENCES

On the rate of convergence to the asymptotic cone for nilpotent groups and subFinsler geometry

Addressing a question of Gromov, we give a rate in Pansu’s theorem about the convergence to the asymptotic cone of a finitely generated nilpotent group equipped with a left-invariant word metric

Stable group theory and approximate subgroups

We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sum-product phenomenon. For a simple linear group G, we show that a finite

Separable and tree-like asymptotic cones of groups

Using methods from nonstandard analysis, we will discuss which metric spaces can be realized as asymptotic cones. Applying the results we will nd in the context of groups, we will prove that a group

The sumset phenomenon

Answering a problem posed by Keisler and Leth, we prove a theorem in non-standard analysis to reveal a phenomenon about sumsets, which says that if two sets A and B are large in terms of measure,

On the Asymptotics of the Growth of 2‐Step Nilpotent Groups

Pansu has shown that the growth function of every virtually nilpotent group Γ with respect to any finite generating set has asymptotics γ(n)∼αnd, where d is the degree of growth of Γ. The paper

The Degree of Polynomial Growth of Finitely Generated Nilpotent Groups

It will be convenient to say that a group G virtually has a property P if some subgroup of finite index has property P. Thus the theorem above concludes that 'G is virtually nilpotent'. (This

Gromov's theorem on groups of polynomial growth and elementary logic

Growth of finitely generated solvable groups and curvature of Riemannian manifolds

If a group Γ is generated by a finite subset 5, then one has the "growth function" gs, where gs(m) is the number of distinct elements of Γ expressible as words of length <m on 5. Roughly speaking, J.

On groups with locally compact asymptotic cones

  • M. Sapir
  • Mathematics
    Int. J. Algebra Comput.
  • 2015
It is shown how a recent result of Hrushovsky implies that if an asymptotic cone of a finitely generated group is locally compact, then the group is virtually nilpotent.