Small cancellation theory over Burnside groups
Joint with Rémi Coulon.
1. A flexible tool for producing infinite periodic groups
Here are possible approaches.
1. Understand the proof that is infinite, and add a few relators to an infinite presentation. Olshanskii’s Tarski monster 1982. Hard.
2. Let be hyperbolic and torsion free. Then has infinite periodic quotients (Olshanskii 1991, Gromov-Delzant). Drawback: the exponent depends on .
3. Our approach.
Theorem 1 (Coulon-Gruber) There exists such that for all , odd, the following holds. Let be a presentation such that
- , no relator of length .
- No relator is a proper power.
- No third power is a subword of a relator.
Then is infinite. Furthermore, no nonempty proper subword of a element of represents the identity in .
2. Examples of admissible data
In a group presentation, a piece is a common subword of two cyclic conjugates of relators. Say the presentation with cyclically reduced relators satisfies if whenever a piece is a subword of a cyclic conjugate of a relator , then .
- In the standard presentation of the genus 2 surface, all pieces have length 1.
- Let . Let be words in and of length . Let contain all words of the form . Then this satisfies .
- Let . Let be the Thue-Morse sequence. It contains no third power as subwords. Let , , , … be the successive length subwords. Apply previous construction, get a presentation. Our theorem provides an infinite periodic group.
For , let . This gives uncountably many different -periodic groups.
1.Get -periodic groups with coarsely embedded expander graphs.
2. Using the existence of -periodic groups whose word problem is unsolvable, we get the following.
Theorem 2 Let be not a prime. Let be a property of groups such that
- There exists a relatively finite presented -periodic group that has .
- There exists a relatively finite presented -periodic group such that any group containing does not have .
Then there does not exist an algorithm that takes a relatively finite presentation of an -periodic group and decides wether it has or not.
Here a relatively finite presentation means that one kills finitely many elements in .
Examples of suitable properties. Triviality, finiteness, being cyclic, abelian, nilpotent, solvable, amenable…
4. Periodic quotients of groups acting acylindrically on hyperbolic spaces
Let be -hyperbolic. Say acts -acylindrically on if for all with , the number of elements of moving each f and at most distance away is at most .
In 2013, Coulon showed that there exists such that for all odd , periodic quotients exist: let act -acylindrically on a -hyperbolic space without elliptics. Then is infinite, and is injective on a ball of radius 3 in the pseudo-metric induced by .
The point is to understand the normal closure of high powers of all loxodromics simultaneously.
The method is small cancellation: implies that is hyperbolic and any cycle labelled by a relator is an isometrically embedded cycle graph. Small overlap implies a linear isometric inequality.
How does this extend to infinite presentations ?
Theorem 3 (Gruber-Sisto) Let be a subset that contains all subwords of elements of . Then is -hyperbolic.
Apply this here. With our assumptions, is nonelementary. Using the assumption that no -st power is a subword of a relator, we show (Abbot and Hume did something similar) that the action of on is -acylindrical, for .
Why ? If is a subword of a relator, then the translation length of on is .