# Non-algebraic quasi-isometric embeddings

What are examples of finitely generated groups $$\Gamma$$ and $$\Lambda$$ such that the metric space $$\Lambda$$ embeds into $$\Gamma$$ quasi-isometrically but such that $$\Lambda$$ is very much not a subgroup of $$\Gamma$$?

Attempt 1 at $$\Lambda$$ is very much not a subgroup of $$\Gamma$$'': $$\Gamma$$ has no subgroup commensurable to $$\Lambda$$.

Edit: the question is answered in the comments by YCor and Ian Agol. More examples are welcome. Specifically, I would be interested in $$\Gamma$$ that have many such $$\Lambda$$ (in particular, further $$\Lambda$$ that are neither QI to $$\Gamma$$ nor $$\mathbb{Z}$$).

Context: in comparing group theory to geometry one often looks at whether homomorphic embeddings are also quasi-isometric embeddings. This question is about the other way around. There are probably tons of examples, the problem is to verify that $$\Lambda$$ is very much not a subgroup of $$\Gamma$$. For example, any two points of the Cantor space $$\partial F_2$$ define a quasi-isometric embedding of $$\mathbb{Z}$$ into $$F_2$$ and only very few (countably many) of these are anywhere close to being homomorphic images, yet $$\Gamma$$ does have a subgroup isomorphic to $$\mathbb{Z}$$.

• (1) Two cocompact lattices in the same simple Lie group of rank $\ge 2$, or rank 1 and superrigid (2) Burger Mozes, or irreducible lattice in $SL_2(Q_p)^2$, vs product of two free groups; similarly, product of two surface groups vs irreducible lattice in $SL_2(R)^2$ (3) lamplighter on $C_p$ and $C_q$ for $p,q$ distinct primes (not QI, but each QI-embeds into the other one), etc. (4) two lattices in the same nilpotent Lie group (a real form having two rational forms: this exists in dim$\ge 6$, for instance this can be found inside the product of two Heisenberg groups); two lattices in SOL. – YCor Oct 22 '18 at 13:39
• (5) every infinite f.g. group $\Gamma$ has a QI-embedded $\mathbf{Z}$, but if $\Gamma$ is torsion (e.g., Golod-Shafarevich, Grigorchuk, Burnside, etc) then it has no subgroup commensurable to $\mathbf{Z}$. Also $\Gamma\wr\Gamma$ then has a QI-embedded $\mathbf{Z}^d$ for every $d\ge 0$, but no subgroup commensurable to $\mathbf{Z}^d$ for any $d\ge 1$. (6) every non-amenable fg group, or virtually solvable group of exponential growth, has a QI-embedded copy of $F_2$, but there's not always a free subgroup (initial examples by Olshanskii; more recent examples by Monod acting on the interval). – YCor Oct 22 '18 at 13:55
• In addition to @YCor's examples, there are hyperbolic planes embedded in sol, which gives quasi-isometric embeddings of surface groups. mathoverflow.net/a/308560/1345 More generally, Fisher-Whyte have explored q.i. embeddings between symmetric spaces. arxiv.org/abs/1407.0445 – Ian Agol Oct 22 '18 at 22:29
• As regards your edit, several of the examples already given have $\Lambda$ neither QI to $\mathbf{Z}$ nor to $\Gamma$. – YCor Oct 23 '18 at 9:56
• I have further examples, but it's a bit endless... – YCor Oct 23 '18 at 11:40