dc.contributor.author Buran, Michal dc.date.accessioned 2020-11-20T15:36:42Z dc.date.available 2020-11-20T15:36:42Z dc.date.submitted 2020-07-24 dc.identifier.uri https://www.repository.cam.ac.uk/handle/1810/313149 dc.description.abstract 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 $\mathcal{G}^c$ is connected if and only if for any infinite index convex-cocompact subgroup $H$ of $W$ and any finite subset $\{ \gamma_1, \ldots , \gamma_n \} \subset W \setminus H$ there is a surjective homomorphism $f$ from $W$ to a finite alternating group such that $f (\gamma_i) \notin f (H)$ . 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 $H_1, \ldots H_k$ 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 $f(H_j)$. dc.description.sponsorship EPSRC International Doctoral Scholar scheme dc.rights All Rights Reserved dc.rights.uri https://www.rioxx.net/licenses/all-rights-reserved/ dc.subject Geometric group theory dc.subject Free groups dc.subject Residual properties dc.subject Probabilistic method dc.title Separability within alternating groups and randomness dc.type Thesis dc.type.qualificationlevel Doctoral dc.type.qualificationname Doctor of Philosophy (PhD) dc.publisher.institution University of Cambridge dc.identifier.doi 10.17863/CAM.60253 rioxxterms.licenseref.uri https://www.rioxx.net/licenses/all-rights-reserved/ rioxxterms.type Thesis dc.publisher.college Trinity dc.type.qualificationtitle PhD in Pure Mathematics cam.supervisor Wilton, Henry cam.supervisor.orcid Wilton, Henry [0000-0001-6369-9478]
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