## Separability within alternating groups and randomness

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## 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

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