Markus Steenbock
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On what finitely generated (left-orderable) simple groups can know about their subgroups Open
In this paper, we survey some of the recent advances on embeddings into finitely generated (left-orderable) simple group such that the overgroup preserves algorithmic, geometric, or algebraic information about the embedded group. We discus…
View article: Random Quotients of Free Products
Random Quotients of Free Products Open
We introduce a density model for random quotients of a free product of finitely generated groups. We prove that a random quotient in this model has the following properties with overwhelming probability: if the density is below $1/2$, the …
Subgroups and diversity of left-orderable small cancellation groups Open
We arrange classical small cancellation constructions to produce left-orderable groups: we show that every finitely generated group is the quotient of a left-ordered small cancellation group by a finitely generated kernel (Rips constructio…
Rips construction without unique product Open
Given a finitely presented group $Q,$ we produce a short exact sequence $1\to N \hookrightarrow G \twoheadrightarrow Q \to 1$ such that $G$ is a torsion-free Gromov hyperbolic group without the unique product property and $N$ is without th…
Product set growth in Burnside groups Open
Given a periodic quotient of a torsion-free hyperbolic group, we provide a fine lower estimate of the growth function of any sub-semi-group. This generalizes results of Razborov and Safin for free groups.
Embeddings into left‐orderable simple groups Open
We prove that every countable left-ordered group embeds into a finitely generated left-ordered simple group. Moreover, if the first group has a computable left-order, then the simple group also has a computable left-order. We also obtain a…
Product set growth in groups and hyperbolic geometry Open
Generalising results of Razborov and Safin, and answering a question of\nButton, we prove that for every hyperbolic group there exists a constant\n$\\alpha >0$ such that for every finite subset $U$ that is not contained in a\nvirtually cyc…
Embeddings into left-orderable simple groups Open
We prove that every countable left-ordered group embeds into a finitely generated left-ordered simple group. Moreover, if the first group has a computable left-order, then the simple group also has a computable left-order. We also obtain a…
A combination theorem for cubulation in small cancellation theory over free products Open
We prove that a group obtained as a quotient of the free product of finitely many cubulable groups by a finite set of relators satisfying the classical –small cancellation condition is cubulable. This yields a new large class of relatively…