The Hilbert-Smith Conjecture (HSC) is a famous open problem in geometric group theory stating "for every prime $p$ there are no faithful continuous action of the $p$-adic group of integers $A_p$ on any connected topological $n$-manifold." It is open (afaik) for all dimensions $n > 3$.

What are some promising directions of research related to the HSC? Specially, what are some realistic (for a beginning grad student) areas of research involving group actions (related/unrelated to HSC) of finite groups?

One example related to the HSC may be trying to investigate what properties must a (hypothetical) faithful $A_p$-action satisfy. Is this direction at all promising? (by promising here I mean, can such properties be 'realistically' derivable? I don't mean if this will work towards solving the HSC.) What are some related questions you would suggest someone investigating further involving group actions from, say, an algebraic point of view? Any relevant papers/preprints one could look at?

The Hilbert-Smith Conjecture (HSC) is a famous open problem in geometric group theory stating "for every prime $p$ there are no faithful continuous action of the $p$-adic group of integers $A_p$ on any connected topological $n$-manifold." It is open (afaik) for all dimensions $n > 3$.

What are some promising directions of research related to the HSC? Specially, what are some realistic (for a beginning grad student) areas of research involving group actions of finite groups?

One example related to the HSC may be trying to investigate what properties must a (hypothetical) faithful $A_p$-action satisfy. Is this direction at all promising? (by promising here I mean, can such properties be 'realistically' derivable? I don't mean if this will work towards solving the HSC.) What are some related questions you would suggest someone investigating further involving group actions from this point of view? Any relevant papers/preprints one could look at?

The Hilbert-Smith Conjecture (HSC) is a famous open problem in geometric group theory stating "there are no faithful continuous action of the $p$-adic group of integers $A_p$ on any connected topological n-manifold." It is open (afaik) for all dimensions $n > 3$.

What are some promising directions of research related to the HSC? Specially, what are some realistic (for a beginning grad student) areas of research involving Group Actions (related/unrelated to HSC) of finite groups?

One example related to the HSC may be trying to investigate what properties must a (hypothetical) faithful $A_p$-action satisfy. Is this direction at all promising? (by promising here I mean, can such properties be 'realistically' derivable? I don't mean if this will work towards solving the HSC.) What are some related questions you would suggest someone investigating further involving group actions from, say, an algebraic point of view? Any relevant papers/preprints one could look at?

The Hilbert-Smith Conjecture (HSC) is a famous open problem in geometric group theory stating "for every prime $p$ there are no faithful continuous action of the $p$-adic group of integers $A_p$ on any connected topological $n$-manifold." It is open (afaik) for all dimensions $n > 3$.

What are some promising directions of research related to the HSC? Specially, what are some realistic (for a beginning grad student) areas of research involving group actions (related/unrelated to HSC) of finite groups?

One example related to the HSC may be trying to investigate what properties must a (hypothetical) faithful $A_p$-action satisfy. Is this direction at all promising? (by promising here I mean, can such properties be 'realistically' derivable? I don't mean if this will work towards solving the HSC.) What are some related questions you would suggest someone investigating further involving group actions from, say, an algebraic point of view? Any relevant papers/preprints one could look at?