Cfrgtkky

I have the following definitions which I want to compare:

Definition 1: Let $G$ be a group and $H$ be a subgroup. $H$ is called normalish if for any finite sequence $g_1,dots,g_nin G$ the intersection $bigcap_n g_nHg_n^{-1}$ is infinite.

Definition 2: Let $G$ be a group and $H$ be a subgroup. $H$ is called recurrent if there exists a finite subset $Fsubset Gsetminus{1}$ such that $Fcap gHg^{-1}neqemptyset$ for all $gin G$.

Or equivalently for Definition 2, if $G$ is discrete, for any sequence $(g_n)_n$ in $G$ there is a subsequence $(g_{n_k})_k$ such that $bigcap_k g_{n_k}Hg_{n_k}^{-1}neq{1}$.

Definition 3: Let $G$ be a group and $H$ be a subgroup. H has the $star$-property if $bigcap_{tin F} tHt^{-1}neq {1}$ for every finite subset $Fsubset G$.
(I don't know if there is a name for such subgroups)

I want to know if these definitions are related, i.e. that from one definitions follow the other. It is clear to me that Def. 2 doesn't imply Def. 1. Furthermore, it is clear that Def. 3 doesn't imply Def. 1.
Could it be that Def. 1 (sometimes) implies Def. 2 and Def. 1 (sometimes) implies Def.3?
(How) are definition 2 and definition 3 related?

Thank you.

To expand on YCor's comment: Of the six possible implications between these properties, only the implication 1$implies$3 holds in general.

This can be easily seen by translating every definition into plain english:

Def. 1="Every finite intersection of conjugates of $H$ is infinite"

Def. 2="Every conjugate of $H$ contains a non-trivial (cyclic) subgroup from a given, finite list".

Def. 3="Every finite intersection of conjugates of $H$ is non-trivial".

Counterexamples:

1.$notRightarrow$2. and 3.$notRightarrow$2.: Consider $G=Sym_{fin}(mathbb{Z})={piin Sym(mathbb{Z}) mid pi(k)=k text{ for all but finitely many } kinmathbb{Z}}$ and $H:={piin G mid pi_{|mathbb{N}}=id}$ the subgroup of elements which only move negative integers. Then ${^{g_1} H}capldotscap{^{g_n} H}={piin G mid pi_{|g_1mathbb{N} cup ldots cup g_n mathbb{N}} = id}$ contains the subgroup which moves only integers $<M$ for some constant $M$ and is therefore infinite. Similarly if $Fsubseteq Gsetminus{1}$ were such that $forall gin G: Fcap{^g H}neqemptyset$, then for all $gin G$ there would exist a $f_gin F$ and some $hin H$ with $Fix(f_g)=Fix(ghg^{-1})=g Fix(h) supseteq gmathbb{N}$. Consider the following sequence $(g_n)$ of permutations
$$g_n(x) := begin{cases} n-x & -nleq xleq 2n \ x &text{otherwise}end{cases}$$
Because $F$ is finite, there exists a subsequence $(g_{n_k})$ such that $f:=f_{g_{n_k}}$ is constant. For this particular $f$ we get $forall k: Fix(f) supseteq g_{n_k}mathbb{N}supseteq g_{n_k}[0,2n_k] = [-n_k,+n_k]$ and therefore $Fix(f)supseteq mathbb{Z}$ so that $f=1$ contrary to the assumption $Fsubseteq Gsetminus{1}$.

2.$notRightarrow$1. and 3.$notRightarrow$1.: For the converse consider $G_0=Aff(mathbb{R})={xmapsto ax+b mid ainmathbb{R}^times, binmathbb{R}}$. This is a Frobenius group with the translation group ${xmapsto x+b}$ as kernel and $H_0:={xmapsto ax}$ as complement, i.e. $H_0cap{^g H_0} = {1}$ for all $gin G_0setminus H_0$. In other words two conjugates are either equal (iff the conjugating elements lie in the same $H$-coset) or intersect trivially. Now let $G:=G_0times X$ and $H:=H_0times X$. Therefore
$${^{g_1} H}capldotscap{^{g_n} H} = begin{cases} H & g_1H=ldots=g_nH \ {1}times X &text{otherwise}end{cases}$$
so that both 2. and 3. hold in this case. If we choose $X$ to be non-trivial but finite, then $H$ is not normalish in $G$.

2.$notRightarrow$3.: Finite groups yield many counterexamples. If $G$ is finite, then every $Hneq 1$ satisfies condition 2 because we can choose $F=Gsetminus{1}$ itself. And $F=G$ is also finite so that 3 claims $core_G(H)=bigcap_{gin G} gHg^{-1} neq 1$ which of course need not be true. Choose for example $G=S_3, H=langle(12)rangle$.