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Separability within alternating groups and randomness 10.17863/CAM.60253 Buran, Michal Wilton, Henry 0000000163699478 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 $f(H_j)$. University of Cambridge 2020-07-24 Thesis Doctoral Doctor of Philosophy (PhD) eng EPSRC International Doctoral Scholar scheme https://www.repository.cam.ac.uk/handle/1810/313149 https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/9ee5efac-dcd7-46c4-800d-7c4a1a78dd29/download 4138db6d6ccbd87589565fe44d108525 https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/dd397889-a4d2-4add-b3ce-45f5044db905/download 353adac0d1ebdfd65ab16480263c3c87 https://www.rioxx.net/licenses/all-rights-reserved/ Geometric group theory Free groups Residual properties Probabilistic method