Monster groups

This post is the result of  my failing to understand Sapir's aspherical Higman embedding theorem. I mostly follow his slides from when he gave talks about his result. It should be fairly accessible given how little I managed to understand. 

 In the world of infinite groups, geometric group theorists tend to study very restricted classes of groups where there is control coming from geometry or analysis because there are all sorts of monsters that lurk in the shadows. However, even the techniques from GGT can be used to produce all sorts of crazy things. The most naive way to do this would simply be add relators to impose crazy phenomenon. This doesn't quite work because there may be unexpected cancellation between group elements, but with a bit more care it produces all manner of exotica.

Definition: Let $G= \braket{S|R}$ be a group presentation where $R \subseteq F(X)$ is a set of freely reduced and cyclically reduced words in the free group $F(X)$ such that $R$ is symmetrised, that is, closed under taking cyclic permutations and inverses.

A nontrivial freely reduced word $u$ in $F(X)$ is called a piece with respect to the presentation if there exist two distinct elements $r_1$, $r_2$ in $R$ that have $u$ as maximal common initial segment.

Note that any presentation can be symmetrized by just adding in the cyclic permutations and inverses without altering the isomorphism type of the group.

Definition: Let $0 < \lambda < 1$. The presentation is said to satisfy the $C'(\lambda)$ small cancellation condition if whenever $u$ is a piece w.r.t. the presentation and $u$ is a subword of some $r \in R$, then $|u| < \lambda |r|$. Here $|v|$ is the length of a word $v$.

There are variants of this condition but the basic idea behind all of them is simply that there shouldn't be too much overlap, and if there is little overlap then many things are preserved in the quotient. 

Consider a tessellation of the hyperbolic plane: 

Since the circumference of a hyperbolic circle grows exponentially in the radius, in order to enclose larger and larger areas we need lots of translates of the fundamental domain and this results in exponential increase in perimeter. The picture also suggests that two intersecting translates of the fundamental domain intersect in segments of small length relative to the perimeter of the fundamental domain, and in fact any loop in the 1-skeleton has to intersect some translate in more than half its perimeter. This gives rise to Dehn's algorithm for the word problem in a surface group: any homotopically trivial word has to have more than half a relator as a subword, so we can use the relator to shorten the word. If, on a given word, the shortening process doesn't terminate at the identity, the word represents nontrivial element. 

Small cancellation groups are a generalisation of surfaces where this same procedure works: the condition on pieces means that overlaps are small, means that trivial words have to have subwords which make up more than half a relator. The fundamental result is:

Greendlinger's lemma: Let (∗) be a group presentation satisfying the C′(λ) small cancellation condition where 0 ≤ λ ≤ 1/6. Let w ∈ F(X) be a nontrivial freely reduced word such that w = 1 in G. Then there is a subword v of w and a defining relator r ∈ R such that v is also a subword of r and such that \[|v|>(1-3\lambda)|r|\]

${\displaystyle \left|v\right|>\left(1-3\lambda \right)\left|r\right|}$

Note that the assumption λ ≤ 1/6 implies that  (1 − 3λ) ≥ 1/2, so that w contains a subword more than a half of some defining relator. When the presentation is finite this gives an algorithm to solve the word problem in linear time, i.e. the Dehn function is linear, so the group is hyperbolic. This is a restatement of the fact that the isoperimetric inequality in this case says that area is linear in perimeter, whereas in Euclidean space it is quadratic.

Importantly, if the presentation satisfies suitable small cancellation conditions and none of the relators are proper powers then the presentation complex is aspherical and the group has cohomological dimension 2. The list of things that have been achieved in GGT with this machinery is too long to list, but today's theme is monsters so I'll elaborate in that direction.

Definition: A Tarski monster group is an infinite group $T$ such that all proper subgroups are cyclic. 

Note that any Tarski monster is generated by any pair of non-commuting elements. One can get similar monsters by demanding abelian rather than cyclic. The first Tarski monsters were constructed by Ol'shanskii by hand, which in fact had a much stronger property. For each prime $p >10^{75}$ he constructed uncountably many non-isomorphic examples where all cyclic subgroups were of order $p$. Note that all of these must be simple. He also constructed torsion-free Tarski monsters. Since his method of choice was intricate small-cancellation theory, the latter are of cohomological dimension 2. Higher-dimensional Tarski monsters have been constructed by Fournier-Facio and Sun. It is open whether there is an amenable Tarski monster, although it is highly suspected not. The existing ones are non-amenable groups with no free subgroups, answering the von-Neumann Day question in the negative.

To construct a torsion-free Tarski monster,

1. Start with a free group $F = \langle x, y \rangle$. List all pairs of words $(u_i, v_i)$ from $F$,
2. Take the first pair $(u_1, v_1)$. If they do not generate the whole group $F$ or a cyclic group, impose two relations $p_1(u_1, v_1) = x$, $q_1(u_1, v_1) = y$. Produce a new group $G_1$.
3. Take the second pair $(u_2, v_2)$. If they do not generate the whole group $G_1$ or a cyclic group, impose two relations $p_2(u_1, v_1) = x$, $q_2(u_1, v_1) = y$. Produce a new group $G_2$.
4. Make sure that $G_2$ is hyperbolic by small-cancellation techniques.
5. Proceed by induction.

The inductive limit $\lim_{i \to \infty} G_i = G$ is a Tarski monster. This presentation is recursive, which wil be important later on. This is an example of the philosophy known sometimes as 'just do it'.

We now discuss another type of monster. Gromov monsters are groups which contain families of expander graphs (i.e. highly connected graphs with relatively few edges), which implies that they cannot coarsely embed into a Hilbert space. They also don't satisfy the Baum-Connes conjecture with coefficients, which is some important and involved operator algebra conjecture, and in fact are the main source of counterexamples to the conjecture. In more detail, let $G_i$ be the Ramanujan expanding sequence of finite graphs i.e. the graphs are $k$-regular for some $k$, girth is increasing, the diameter is approximately the girth, the rank of the fundamental group of $G_i$ is not too large and the second eigenvalues of the incidence matrices are bounded away from the first eigenvalue. Then there exists a finitely generated group whose Cayley graph contains a (coarse) copy of $\sqcup G_i$, which we call a Gromov monster.

In Gromov's original construction the graphs aren't coarsely embedded, but Osajda later gave graphical small-cancellation examples where the graphs are isometrically embedded. Here's how to build one:

To construct a Gromov monster,

1.  Start with the free group $F_k = \langle x_1, \ldots, x_k \rangle$.
2. Pick the first graph in the expander sequence, $G_1$.
3. Consider a random labeling of edges of $G_1$ by letters $x_1^{\pm 1}, \ldots, x_k^{\pm 1}$.
4. For every loop $p$ of a generating set of the fundamental group $\pi_1(G_1)$ impose relation $\text{label}(p) = 1$.
5. Make sure that the resulting group is hyperbolic (that is true with probability $> 0$).
6. Proceed by induction, choosing the next graph from the expanding sequence with large enough girth.

The inductive limit $\lim_{G_i} G_i$ is a Gromov monster. Note that the presentation is recursive. One relevant result for maintaining hyperbolicity is the next theorem, due to Coulon.

Let $G$ be a torsion free, non-elementary $\delta$-hyperbolic group generated by $S$.Let $\theta$ be a graph labelled by $S \cup S^{-1}$, $T$ its universal cover and $f : T \to \mathrm{Cay}(G)$ the map given by the labeling. $\Delta$ and $\rho$ are the small cancellation parameters associated to the pair $(f(T), \pi_1(\theta))$. We assume that $\pi_1(\theta) \subset G$ does not contain a proper power.

Theorem 1: Let $\alpha > 1$. There exists positive numbers $\varepsilon$, and $K$ which only depend on $\alpha$ with the following property. Let $\beta > 0$. Assume that $f$ is a $(\alpha, \beta, \frac{1}{2} \mathrm{girth}(\theta))$-local quasi-isometry such that \[\frac{\delta}{\rho}, \frac{\beta}{\rho}, \frac{\Delta}{\rho} \leq \varepsilon.\]

Then  

• $\bar{G} = G / \langle \langle \pi_1(\theta) \rangle \rangle$ is torsion-free, non-elementary, word-hyperbolic. Its hyperbolicity constant only depends on $\delta$, $\alpha$, $\beta$, $\Delta$ and $\rho$.  
• The map $G \to \bar{G}$ induces an isometry from $B(1, K\rho)$ onto its image.

Now here's the kicker: there exists a finitely generated group that is both Tarski monster and Gromov monster. I found this rather shocking, but once the shock subsided it was not too hard to see how to build such a hybrid: the main idea is to alternate the steps of the construction. 

Not being content with crazy groups, however, we can go further and create crazy manifolds using the following remarkable theorem of Sapir:

Theorem 2: Every recursively presented finitely generated group with 2-dimensional $K(\cdot,1)$ embeds into a finitely presented group with finite 2-dimensional $K(\cdot,1)$.

Recall that Higman's embedding theorem is the classical result which says the same thing except without '2-dimensional' in the hypothesis and conclusion. Being able to preserve asphericity is an incredible improvement.

Corollary 3: There exists a closed compact Riemannian aspherical 5-manifold $M^5$ such that the universal cover $\tilde{M}^5$

•  contains an expander,
• has infinite asymptotic dimension,
• does not coarsely embed into a Hilbert space,
• does not satisfy the Baum-Connes conjecture with coefficients, 
• admits a free action by a Tarski monster (and much more).

All of these are consequences of having a subgroup which is both a Tarski and Gromov monster. In dimensions at least 4, there are also truly monstrous manifolds out there.

To deduce the corollary, one uses the reflection group trick. The reader is referred to Davis' book for details, but briefly here's how it works.

A finite 2-dimensional $K(G,1)$ always embeds into $\mathbb{R}^5$. Let $M^5$ be its regular neighbourhood in $\mathbb{R}^5$. Triangulate the boundary.

Consider the Coxeter group $C$ with (right angled) Coxeter graph the 1-skeleton of the triangulation.

Take the factor
\[U = C \times M^5 / \sim\]
where $(g, x) \sim (1, x)$ for a generator $g$ if $x$ is in the closed star of the vertex $g$ in the barycentric subdivision. It is aspherical, open, admits a co-compact action of $C$. Take a torsion-free subgroup $H < C$ of finite index. The manifold $U/H$ is compact, closed and aspherical, $\pi_1(U/H)$ contains $G$.