Separability within alternating groups and randomness

This thesis promotes known residual properties of free groups, surface groups, right angled Coxeter groups and right angled Artin groups to the situation where the quotient is only allowed to be an alternating group. The proofs follow two related threads of ideas. The first thread leads to `alternating' analogues of extended residual finiteness in surface groups \cite{scott1978subgroups}, right angled Artin groups and right angled Coxeter groups \cite{haglund2008finite}. Let $W$ be a right-angled Coxeter group corresponding to a finite non-discrete graph $\mathcal{G}$ with at least $3$ vertices. Our main theorem says that \mathcal{G}<sup>c</sup> is connected if and only if for any infinite index convex-cocompact subgroup $H$ of $W$ and any finite subset \{ γ<sub>1</sub>, \ldots , γ<sub>n</sub> \} \subset W \setminus H there is a surjective homomorphism $f$ from $W$ to a finite alternating group such that f (γ<sub>i</sub>) \notin f (H) . A corollary is that a right-angled Artin group splits as a direct product of cyclic groups and groups with many alternating quotients in the above sense. Similarly, finitely generated subgroups of closed, orientable, hyperbolic surface groups can be separated from finitely many elements in an alternating quotient, answering positively a conjecture of Wilton \cite{wilton2012alternating}. The second thread uses probabilistic methods to provide `alternating' analogues of subgroup conjugacy separability and subgroup into-conjugacy separability in free groups \cite{bogopolski2010subgroup}. Suppose H₁, \ldots H_k are infinite index, finitely generated subgroups of a non-abelian free group $F$. Then there exists a surjective homomorphism f:F \longrightarrow A_m such that if H_i is not conjugate into H_j, then f(H_i) is not conjugate into f(H_j).

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2020

Author dc:creator

• Buran, Michal

Advisor dc:contributor.advisor

• Wilton, Henry

• Geometric group theory
• Free groups
• Residual properties
• Probabilistic method

Licence

• https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/dd397889-a4d2-4add-b3ce-45f5044db905/download
• https://www.rioxx.net/licenses/all-rights-reserved/

Language dc:language

eng

Author Identifier

0000-0001-6369-9478

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/313149