Dtjytyk

Let $G$ be a group and $H unlhd G$. In general, $H=H_Z sqcup H_{G setminus Z}$, where $H_Z:=H cap Z(G)$ and $H_{G setminus Z}:=H cap (G setminus Z(G))$. I'm investigating on a plausible visual model for the pair $(G,H)$. I'll provisionally retain the following one, till some inconsistency will pop up for revision/reject. By "inconsistency" I mean either a contradiction with, or the inability to show, known algebraic facts.

1. 
   $G$ is the euclidean 3-space and $e$ its (geometrical) center;


2. given $g in G setminus Z(G)$, the centralizer $C_G(g)$ is a ball whose poles are $g$ and the element $g_{operatorname{op}}$, opposite to $g$ with respect to $e$ and distant $mathtt{r}_Z$ from $e$;


3. by 2, $Z(G)=bigcap_{g in G}C_G(g)$ is the ball centered in $e$ of radius $mathtt{r}_Z$;
   
   


4. given $g in G$, the right cosets $C_G(g)g'$, $g' in G$, are eccentric, thick "shells" embedding $C_G(g)$ ("onion"-like); the eccentricity gives the possibility to single out as another partition of $G$ the one made of the left cosets; shell's average thickness is a decresing function of shell's size, so as to get a hint of the bijection between any pair of cosets (constant volume);
   
   


5. 
   $forall h in H_Z$, the conjugacy orbit by $h$ is pointwise, being $O_h=lbrace g^{-1}hg, g in G rbrace = lbrace h rbrace$;


6. by 5, $H_Z$ is an axis of $Z(G)$ (or anything topologically equivalent to that);


7. once popped out of $Z(G)$, conjugacy orbits become real ones, namely circles around the axis induced by $H_Z$, which globally form a "polar" toroidal surface, embedding $Z(G)$ (this is $H_{G setminus Z}$);


8. 
   $H$ splits $G setminus H$ into two regions: an "inner" one and an "outer" one, say $G setminus H = G_{<H} sqcup G_{>H}$; given $g in G_{<H}$, the coset $Hg$ is the toroidal surface by $g$, slicing $Z(G)$; given $g' in G_{>H}$, the coset $Hg'$ is the surface by $g'$, embedding $H$ and topologically equivalent to a 2-sphere.
   
   

This model is possibly still far from the "reality", but maybe we can get a better one by addressing some points raised by this one:

#1. Does $H$'s closure have some algebraic validity? What would it mean?

#2. Would the special case $Z(G)=lbrace e rbrace$ be consistently described by the above model, i.e. with $Z(G)$ "deflated" down to one point?

#3. Given $h in H_{G setminus Z}$, are the algebraic loci $C_G(h) cap H$ and $C_G(h) cap O_h$ suitably accounted for in terms of the sphere/torus crossing expected from the model?

I am reading the question as you are asking if there is a group $G$ and a subgroup $H$ that satisfy the above properties. I haven't read through all of your properties, but it is already impossible with properties 2, 3.

Namely, say you have a group $G$ which can be identified with $mathbb R^3$. Your property 3 says $Z(G)$ is a ball of some radius, say $r_Z$, about $e$. (You say sphere, but I assume you mean ball, because certainly you need $e in Z(G)$. I will assume the same for property 2.)

Pick any $g in G - Z(G)$. Then property 2 says $C_G(g)$ is a ball of some radius $r > r_e$ about $e$. Take some $h$ in $C_G(g) - Z(G)$ in the interior of this ball. Then $C_G(h)$ is a ball of radius $< r$ about $e$, but $gh=hg$ means $g in C_g(h)$. However $g$ has distance $r$ from $e$, a contradiction.