Embedding complexes and group actions on manifolds

January 17, 2026

Embedding complexes and group actions on manifolds

It has long been of interest to mathematicians when geometric objects can embed or immerse into Euclidean space. The first instance of this is the consideration of planar graphs (which are 1-dimensional simplicial complexes): Kuratowski showed that a graph is planar if and only if it doesn't contain a subgraph which is isomorphic to a subdivided copy of the complete graph $K_5$ or the utilities graph $K_{3,3}$. See also the later theorem of Wagner which says that a planar graph is precisely one which doesn't contain either of these as a minor.

Next came Whitney, whose celebrated theorem says that any $n$-dimensional topological manifold embeds in $\mathbb{R}^{2n}$ and immerses into $\mathbb{R}^{2n-1}$. The dimensions are sharp: the Klein bottle doesn't embed in $\mathbb{R}^3$. See here for a discussion about different ways to show the non-embeddability of the Klein bottle.

A natural generalisation of this is a less well-known result of Stallings that any $n$-dimensional complex is homotopic to one which embeds in $\mathbb{R}^{2n}$. This post discusses a paper of Bestvina-Kapovich-Kleiner that shows this is sharp as a byproduct of a more general result about groups acting properly discontinuously on contractible manifolds: groups have a natural notion of dimension on them and one would hope that in well-behaved groups they cannot admit too wild actions on manifolds of smaller dimension.

van Kampen's work

We begin by recalling earlier work of van Kampen on obstructions to embeddability. Cohomology is taken with $\mathbb{Z}/2\mathbb{Z}$ coefficients because this is how one detects orientability. Indeed, the Klein bottle example shows that this is an important consideration since the compact orientable surfaces embed in $\mathbb{R}^3$.

$\textbf{Definition:}$ Fix a non-negative integer $m$. A finite simplicial complex $K$ of dimension $\leq m$ is an $m$-obstructor complex if the following holds:

There is a collection \[ \Sigma=\big\{\{\sigma_{i},\tau_{i}\}_{i=1}^{k}\big{\}}\] of unordered pairs of disjoint simplices of $K$ with $\dim\sigma_{i}+\dim\tau_{i}=m$ that determine an $m$-cycle (over $\mathbb{Z}/2\mathbb{Z}$) in \[ \bigcup\{\sigma\times\tau\subset K\times K|\sigma\cap\tau=\emptyset\}/(\mathbb{Z}/2\mathbb{Z})\] where $\mathbb{Z}/2\mathbb{Z}$ acts by $(x,y)\mapsto(y,x)$.
For some (any) general position map $f:K\to\mathbb{R}^{m}$ the (finite) number \[ \sum_{i=1}^{k}|f(\sigma_{i})\cap f(\tau_{i})|\] is odd.
For every $m$-simplex $\sigma\in K$ the number of vertices $v$ such that the unordered pair $\{\sigma,v\}$ is in $\Sigma$ is even.

It turns out van Kampen's obstruction is the only obstruction to the existence of embeddings of complexes of dimension $\geq 3$ into $\mathbb{R}^{2n}$. For 2-dimensional complexes there are other obstructions as well.

Let $K$ be an $m$-obstructor complex. If $f$ and $f^{\prime}$ are two general position maps $K\to\mathbb{R}^{m}$ choose a general position homotopy $H$ between them. If one thinks of the homotopy as a continuous deformation between two maps parametrised by the interval $[0,1]$, the 'time parameter', 'watching it evolve' shows that in the presence of item 1 the two integers from item 2 for $f$ and $f^{\prime}$ differ by an even integer. In particular, item 2 implies that $K$ does not embed in $\mathbb{R}^{m}$; indeed for every map $K\to\mathbb{R}^{m}$ there exist two disjoint simplices of $K$ whose images intersect.

We will view $\Sigma$ as a subcomplex of $\bigcup\{\sigma\times\tau\subset K\times K|\sigma\cap\tau=\emptyset\}/(\mathbb{Z}/2\mathbb{Z})$. Then item 1 states that $\Sigma$ is a $m$-pseudomanifold over $\mathbb{Z}/2\mathbb{Z}$, meaning that every $(m-1)$-cell is the face of an even number of $m$-cells, and in particular we have the fundamental class $[\Sigma]\in H_{m}(\Sigma)$. Similarly, the collection $\tilde{\Sigma}$ of \textit{ordered} pairs corresponding to the pairs in $\Sigma$ can be viewed as a subcomplex of $\bigcup\{\sigma\times\tau\subset K\times K|\sigma\cap\tau=\emptyset\}$ and is an $m$-pseudomanifold. Further, $(x,y)\mapsto(y,x)$ is the deck transformation of the natural double cover $\widetilde{\Sigma}\to\Sigma$. Let $\phi:\Sigma\to\mathbb{R}P^{\infty}$ be a classifying map for this double cover. We note that item 2 is equivalent to the requirement that

\[\langle\phi^{*}(w^{m}),[\Sigma]\rangle\neq 0\in\mathbb{Z}/2\mathbb{Z}\]

where $w^{m}\in H^{m}(\mathbb{R}P^{\infty})$ is the nonzero class. Indeed, we can perturb $f:K\to\mathbb{R}^{m}$ to a map $F=(f,g):K\to\mathbb{R}^{m}\times\mathbb{R}=\mathbb{R}^{m+1}$ so that $F(\sigma_{i})\cap F(x)=\emptyset$ for all $i$. Then we have a classifying map $\phi:\Sigma\to\mathbb{R}P^{m}\subset\mathbb{R}P^{\infty}$ defined by

\[\phi((x,y))=\text{line through }F(x)\text{ and }F(y)\]

where a point of $\mathbb{R}P^{m}$ is viewed as the set of parallel lines in $\mathbb{R}^{m}$. Then $\langle\phi^{*}(w^{m}),[\Sigma]\rangle$ can be computed as the "degree" of $\phi$, which in turn is the number of points of $\Sigma$ mapped to the "vertical lines" $pt\times\mathbb{R}$, i.e., the number from item 2.

A restatement of item 3 is that the projection map $\pi:\widetilde{\Sigma}\to K$ (say to the second coordinate) has the property that the pullback of every $m$-cocycle evaluates trivially on the fundamental class $[\widetilde{\Sigma}]$.

We impose item 3 to ensure that two useful facts hold. The first is

$\textbf{Linking Lemma:}$ Suppose $ W $ is a contractible $(m+1)$-manifold, $ K $ is an $ m $-obstructor complex and $ G : K \times [0, \infty) \to W $ is a (continuous) proper map. Then there exist two disjoint simplices $ \sigma, \tau $ in $ K $ such that $ G(\sigma \times \{0\}) \cap G(\tau \times [0, \infty)) \neq \emptyset $.

We will need this later. In the simplest instance, the linking lemma states that the utilities graph embedded in $\mathbb{R}^3$ links every push-off of itself.

The second fact which requires item 3 is the following, which allows us to produce lots and lots of obstruction complexes:

$\textbf{The Join Lemma}$ If $ K_j $ is an $ m_j $-obstructor complex for $ j = 1,2 $ then the join $ K_1 * K_2 $ is an $ (m_1 + m_2 + 2) $-obstructor complex.

We record here another way of producing obstructor complexes:

$\textbf{Cone Lemma}$ If $ K $ is an $ m $-obstructor complex, then the cone $ cK $ is an $ (m + 1) $-obstructor complex.

Van Kampen's obstruction theory can be summarized in the following proposition:

$\textbf{Proposition 1:}$ Suppose that $K$ is an $m$-obstructor complex and that $W$ is a contractible $m$-manifold. Then for every map $F:K\to W$ there exist disjoint simplices $\sigma$ and $\tau$ in $K$ such that $F(\sigma)\cap F(\tau)\neq\emptyset$. In particular, $K$ does not embed into $W$.

To prove this we will need the following:

$\textbf{Lemma 2.}$ Suppose $W$ is a contractible manifold of dimension $m$. Then the space $W\times W\setminus\Delta/(\mathbb{Z}/2\mathbb{Z})$ of unordered pairs of points in $W$ is homotopy equivalent to $\mathbb{R}P^{m-1}$.

$\textbf{Proof:}$ We may assume that $n>2$ since otherwise $W$ is homeomorphic to $\mathbb{R}^{n}$. Let $U\subset W$ be a (small) open set homeomorphic to $\mathbb{R}^{n}$. Consider the diagram

Note that $ U \times U \setminus \Delta $ fibers over $ U $ with fiber $ U \setminus pt \simeq S^{n-1} $; thus $ U \times U \setminus \Delta \simeq S^{n-1} $ and similarly $ W \times W \setminus \Delta \simeq S^{n-1} $; moreover, inclusion
\[U \times U \setminus \Delta \hookrightarrow W \times W \setminus \Delta\]
is a homotopy equivalence. Since for $ n > 2 $ the two spaces in the first row of the above diagram are simply-connected, it follows that
\[(U \times U \setminus \Delta)/(\mathbb{Z}/2\mathbb{Z}) \hookrightarrow (W \times W \setminus \Delta)/(\mathbb{Z}/2\mathbb{Z})\]
induces an isomorphism in homotopy groups, and is therefore a homotopy equivalence. $\blacksquare$

$\textbf{Proof of Proposition 1:}$ The case of $W=\mathbb{R}^{m}$ was discussed above. For the general case, assume on the contrary that $F:K\to W$ violates the proposition. Define $\phi:\Sigma\to W\times W\setminus\Delta/(\mathbb{Z}/2\mathbb{Z})$ by $\phi((x,y))=\{F(x),F(y)\}$. The following lemma then implies that $\Sigma$ classifies into $\mathbb{R}P^{m-1}$, a contradiction. $\blacksquare$

Actions on manifolds

Recall that a (continuous) map $h:A\to B$ is $\textit{proper}$ if the preimages of compact sets are compact. We say that maps $h_{1}:A_{1}\to B$ and $h_{2}:A_{2}\to B$ into a metric space $B$ $\textit{diverge}$ (from each other) if for every $D>0$ there are compact sets $C_{i}\subset A_{i}$ such that $h_{1}(A_{1}\setminus C_{1})$ and $h_{2}(A_{2}\setminus C_{2})$ are $>D$ apart. If $K$ is a finite complex, we define the open cone $cone(K)=K\times[0,\infty)/K\times\{0\}$. If $K$ is also an obstructor complex, we say that a proper map $h:cone(K)\to B$ is $\textit{expanding}$ if for disjoint simplices $\sigma$, $\tau$ in $K$ the maps $h|cone(\sigma)$ and $h|cone(\tau)$ diverge. It will also be convenient to make the analogous definition on the level of $0$-skeleta. Triangulate $cone(K)$ so that $cone(\sigma)$ is a subcomplex whenever $\sigma$ is a simplex of $K$. We say that a proper map $h:cone(K)^{(0)}\to B$ is \textit{expanding} if for all pairs $\sigma$, $\tau$ of disjoint simplices in $K$ the restrictions $h|cone(\sigma)^{(0)}$ and $h|cone(\tau)^{(0)}$ diverge. We also equip $cone(K)^{(0)}$ with the edge-path metric, so that a map $h:cone(K)^{(0)}\to B$ is $\textit{Lipschitz}$ if there is a uniform upper bound on the distance between the images of adjacent vertices in $cone(K)^{(0)}$.

Note that if $h:cone(K)\to B$ is a proper expanding map, then there is $t_{0}\geq 0$ such that the map $G:K\times[0,\infty)\to B$ defined by $G(x,t)=h([x,t+t_{0}])$ satisfies $G(\sigma\times\{0\})\cap G(\tau\times[0,\infty))=\emptyset$ for any two disjoint simplices $\sigma$, $\tau$ of $K$.

A proper map $h:A\to B$ between proper metric spaces is $\textit{uniformly proper}$ if there is a proper function $\phi:[0,\infty)\to[0,\infty)$ such that

\[d_{B}(h(x),h(y))\geq\phi(d_{A}(x,y))\]

for all $x$, $y\in A$. This notion is weaker than the notion of a quasi-isometric embedding, which would require $\phi$ to be a linear function.

Let $\Gamma$ be a finitely generated group equipped with the word-metric with respect to some finite generating set. We now define a few different notions of dimension of $\Gamma$.

$\textbf{Definition:}$ The $\textit{obstructor dimension}$ $\operatorname{obdim}(\Gamma)$ is defined to be $0$ for finite groups, $1$ for $2$-ended groups, and otherwise $m+2$ where $m$ is the largest integer such that for some $m$-obstructor complex $K$ and some triangulation of the open cone $cone(K)$ as above there exists a proper, Lipschitz, expanding map $f:cone(K)^{(0)}\to\Gamma$. If no maximal $m$ exists we set $\operatorname{obdim}(\Gamma)=\infty$.

$\textbf{Remark:}$ Clearly, one can replace $\Gamma$ in the above definition by any quasi-isometric proper metric space. In particular, if $\Gamma$ acts cocompactly, properly discontinuously, and isometrically on a proper geodesic metric space $X$, we can substitute $X$ for $\Gamma$. Moreover, if $\Gamma$ is of type $F_{m+1}$ so that $X$ can be chosen to be $m$-connected, then $f$ can be extended to a proper, expanding map $\tilde{f}:cone(K)\to X$ with a uniform bound on the diameter of the image of any simplex. One advantage of having the (continuous) map defined on the whole cone is that the requirement that the map be Lipschitz can be dropped: one can always triangulate $cone(K)$ to make the same map Lipschitz.

Note that if $\Gamma$ is infinite and not $2$-ended, then we can take $K$ to consist of $3$ points, so $\operatorname{obdim}(\Gamma)\geq 2$.

$\textbf{Definition:}$ The $\textit{uniformly proper dimension}$$\operatorname{updim}(\Gamma)$ is the smallest integer $n$ such that there is a contractible $n$-manifold $W$ equipped with a proper metric $d_{W}$ so that there is a Lipschitz, uniformly proper map $g:\Gamma\to W$ and so that in addition there is a \textit{contractibility function} $\rho:(0,\infty)\to(0,\infty)$ such that any ball of radius $r$ centered at a point of the image of $g$ is contractible in the ball of radius $\rho(r)$ centered at the same point. If no such $n$ exists we set $\operatorname{updim}(\Gamma)=\infty$.

One usually requires of the contractibility function that the statement about balls be true regardless of where the center is. If we omit the requirement altogether, the invariant would be trivial: every finitely generated group admits a uniformly proper map into $[0,\infty)$. Just choose an injective map $g:\Gamma\to\mathbb{N}\subset[0,\infty)$. The largest metric on $[0,\infty)$ that makes $g$ 1-Lipschitz and makes all $[n,n+1]$ isometric to a standard closed interval (of length dependent on $n$) is proper. Of course, this metric is not a path-metric, but insisting on path-metrics would only raise the dimension by $1$: For every $\Gamma$ there is a proper path-metric on $\mathbb{R}^{2}$ and a uniformly proper map $\Gamma\to\mathbb{R}^{2}$.

$\textbf{Definition:}$ The $\textit{action dimension}$ $\operatorname{actdim}(\Gamma)$ is the smallest integer $n$ such that $\Gamma$ admits a properly discontinuous action on a contractible $n$-manifold. If no such $n$ exists, then $\operatorname{actdim}(\Gamma)=\infty$.

Denote also by $\operatorname{gdim}(\Gamma)$ the $\textit{geometric dimension}$ of $\Gamma$, i.e., the minimal $n$ such that $\Gamma$ admits a properly discontinuous action on a contractible $n$-complex. Recall that for virtually torsion-free groups $\Gamma$, $\operatorname{gdim}(\Gamma)$ the Eilenberg-Ganea conjecture states that this is equal to the virtual cohomological dimension $\operatorname{vcdim}$ of $\Gamma$. It is known that the only potential counterexamples would have $\operatorname{gdim}=3$ and $\operatorname{vcdim}=2$, but at least one of the Eilenberg-Ganea conjecture and the Whitehead conjecture have to be false (originally proved by Bestvina and Brady using combinatorial Morse theory).

We note that $\operatorname{updim}(\Gamma)\leq\operatorname{actdim}(\Gamma)$ by choosing a proper invariant metric on $W$ and taking an orbit of the action, and that for torsion-free groups $\Gamma$ we have $\operatorname{actdim}(\Gamma)\leq 2\cdot\operatorname{gdim}(\Gamma)$ by the Stallings theorem cited in the introduction. Alternatively, we could find a $(2n)$-dimensional thickening of an $n$-complex by immersing it in $\mathbb{R}^{2n}$ and taking a regular neighborhood. The inequality $\operatorname{actdim}(\Gamma)\leq 2\cdot\operatorname{gdim}(\Gamma)$ is false for groups with torsion; indeed, the free product $A_{5}*A_{5}$ acts properly discontinuously on a tree but not on the plane. On the other hand, $\Gamma=A_{5}*A_{5}$ contains a free subgroup $\Gamma^{\prime}$ of finite index, hence $2=\operatorname{actdim}(\Gamma^{\prime})<\operatorname{actdim}(\Gamma)$.

The main theorem is:

$\textbf{Theorem 3:} $$\operatorname{obdim}(\Gamma)\leq\operatorname{updim}(\Gamma)$.

$\textbf{Proof:}$ The special cases when $\operatorname{obdim}(\Gamma)\leq 1$ are clear. Let $K$ be an $m$-obstructor complex and $f:cone(K)^{(0)}\to\Gamma$ a proper, Lipschitz, expanding map. Let $W$ be a contractible $n$-manifold with a proper metric and $g:\Gamma\to W$ a uniformly proper Lipschitz map satisfying the contractibility function requirement. Consider the composition $gf:cone(K)^{(0)}\to W$. Now extend $gf$ inductively over the skeleta of $cone(K)$ to get a map $G:cone(K)\to W$. Using the contractibility function, we can arrange that the diameter of the image of each simplex of $cone(K)$ is uniformly bounded. It follows that $G$ is a proper expanding map, and therefore $n\geq m+2$ by the Linking Lemma.

This theorem immediately implies the following chain of inequalities (with the last inequality only for torsion-free groups):

\[\operatorname{obdim}(\Gamma)\leq\operatorname{updim}(\Gamma)\leq \operatorname{actdim}(\Gamma)\leq 2\operatorname{gdim}(\Gamma) \]

The second inequality can be strict. The Baumslag-Solitar group

\[
B=\langle x,t|xt=t^{2}x\rangle
\]

is not a 3-manifold group and so $\operatorname{actdim}(B)=2\cdot\operatorname{gdim}(B)=4$. On the other hand, $\operatorname{obdim}(B)=\operatorname{updim}(B)=3$. The group $B$ admits a uniformly proper map into $\mathbb{H}^{3}$ and the universal cover of the presentation 2-complex admits a proper expanding map defined on the open cone on the tripod, which is a 1-obstructor complex. All three invariants in (1) are monotone, in the sense that if $\Gamma^{\prime}$ is a finitely generated subgroup of $\Gamma$, then $dim(\Gamma^{\prime})\leq dim(\Gamma)$ for any of the above notions of dimension. We also note that both obdim and updim are invariant under quasi-isometries. This is not the case for actdim (even for torsion-free groups) as there are examples of torsion-free groups that are not 3-manifold groups but contain 3-manifold groups as finite index subgroups.

$\textbf{Corollary 5:}$ Let $X = X_1 \times \cdots X_n$ be the $n$-fold product of connected graphs $X_i$ whose fundamental group is the free group $F_2$. Then $X$ does not immerse up to homotopy into $\mathbb{R}^{2n-1}$.

This shows that the result Stallings mentioned earlier is sharp.

$\textbf{Proof:}$ Suppose $X$ is homotopic to a complex $Y$ that immerses in $\mathbb{R}^{2n-1}$. Then Y has a regular neighborhood in $\mathbb{R}^{2n-1}$ (also known as a thickening) which is an aspherical $(2n - 1)$-manifold with fundamental group $F_2 ^n$. But then the universal covering action violates Theorem 4. $\blacksquare$

Short exact sequences

A natural question to ask is now how the various notions behave under extensions of groups. In the case of direct products we have

$\textbf{Lemma 5:}$
\[\operatorname{obdim}(\Gamma_{1}\times\Gamma_{2})\geq\operatorname{obdim}(\Gamma _{1})+\operatorname{obdim}(\Gamma_{2})\]
while
\[\operatorname{updim}(\Gamma_{1}\times\Gamma_{2})\leq\operatorname{updim}(\Gamma _{1})+\operatorname{updim}(\Gamma_{2})\]
and
\[\operatorname{actdim}(\Gamma_{1}\times\Gamma_{2})\leq\operatorname{actdim}(\Gamma _{1})+\operatorname{actdim}(\Gamma_{2}).\]

When one knows more about the geometry/topology of the groups more can be said. Yoon, a student of Bestvina, showed that if $M_i$ is a compact aspherical $n_i$-manifold with all boundary components aspherical and incompressible, and $M_i$ is not homotopy equivalent to an $(n_i-1)$-manifold, then the action dimension of G = $\pi_1(M_1) \times \cdots \times \pi_1(M_k)$ is $n_1 + \cdots + n_k$.

Now suppose only that a group $G$ fits in a short exact sequence

\[1\to H\to G\stackrel{{\pi}}{{\to}}Q\to 1\]

where all groups are finitely generated. The natural guess is that

\[\operatorname{obdim}G\geq\operatorname{obdim}H+\operatorname{obdim}Q \]

and this is true under certain technical assumptions on $\pi$ (admits a Lipschitz section) and $H$ (weakly convex). (As is customary for GGT, all groups are equipped with word metrics.) We note that $\textit{some}$ restrictions are clearly necessary. For instance, the Rips construction gives examples of epimorphisms $G\to Q$ where $G$ is a 2-dimensional hyperbolic group and $Q$ is a prescribed finitely generated group, so that the kernel $H$ is finitely generated (and is neither finite nor 2-ended) and $G$ can be assumed to have Menger curve boundary. Then $\operatorname{obdim}(G)=4$, $\operatorname{obdim}H\geq 2$ and $Q$ can be chosen to have $\operatorname{obdim}(Q)$ as large as one likes.

$\textbf{Definition:}$ We say that a finitely generated group $\Gamma$ is \textit{weakly convex} if there is a collection of (discontinuous, of course) paths $\{\phi_{z,w}:[0,1]\to\Gamma\}_{z,w\in\Gamma}$ and a constant $M>0$ satisfying the following properties:

$\phi_{z,w}(0)=z$ and $\phi_{z,w}(1)=w$.
There is a function $\gamma:[0,\infty)\to[0,\infty)$ such that \[ d(z,w)\leq R\Longrightarrow\operatorname{diam}(Im(\phi_{z,w}))\leq\gamma(R).\]
For all $z,w\in\Gamma$ there is $\epsilon>0$ such that $\phi_{z,w}$ sends subintervals of length $<\epsilon$ to sets of diameter $<M$.
If $d(z,z^{\prime})\leq 1$ and $d(w,w^{\prime})\leq 1$ then for all $t\in[0,1]$ \[d(\phi_{z,w}(t),\phi_{z^{\prime},w^{\prime}}(t))\leq M. \]

The paths are to be thought of as being piecewise constant. We could avoid talking about discontinuous functions by requiring that they be defined only on the rationals in $[0,1]$. It is more standard to think of paths in $\Gamma$ as eventually constant 1-Lipschitz functions defined on non-negative integers; however, for what follows it is important that all paths be defined on the same bounded set. It is possible to reparametrize such paths by "constant speed" paths defined on $[0,1]$.
The collection of paths as above is usually called a "combing" (except for the domain being $[0,1]$). Condition 2 is then a weak version of the requirement that the combing be quasi-geodesic and it follows automatically if the combing is equivariant. Condition 3 is the replacement of the 1-Lipschitz requirement. Condition 4 is the "Fellow Traveller" property.
If $\Gamma$ and $\Gamma^{\prime}$ are quasi-isometric and one is weakly convex, so is the other. Hyperbolic groups, $CAT(0)$ groups, and semi-hyperbolic groups are weakly convex by work of Alonson and Bridson.

We can regard the given paths in the definition of weak convexity as a recipe for extending maps into $\Gamma$ defined on the (ordered) vertices of a 1-simplex to the whole 1-simplex. It is easy to see that one can similarly extend maps defined on the vertices of an $n$-simplex for any $n>0$, with the constant $M=M(n)$ above depending on $n$. By

\[\Delta^{n}=\left\{(t_{0},t_{1},\cdots,t_{n})\in\mathbb{R}^{n+1}|t_{i}\geq 0,t_{0} +t_{1}+\cdots+t_{n}=1\right\}\]

we denote the standard $n$-simplex, and by $I_{n,k}$ the standard face inclusion $\Delta^{n-1}\hookrightarrow\Delta^{n}$ onto the face $t_{k}=0$ given by

\[I_{n,k}(t_{0},t_{1},\cdots,t_{n-1})=(t_{0},t_{1},\cdots,0,\cdots,t_{n-1}).\]

We can now state the theorem for short exact sequences in general:
$\textbf{Theorem 6:}$ Let

\[1\to H\to G\stackrel{{\pi}}{{\to}}Q\to 1\]

be a short exact sequence of finitely generated groups. Suppose that $H$ is weakly convex and that $\pi$ admits a Lipschitz section $s:Q\to G$. Then

\[\operatorname{obdim}G\geq\operatorname{obdim}H+\operatorname{obdim}Q.\]

We refer the interested reader to the original paper for the proof.

$\textbf{Corollary 7:}$ If $G=H\rtimes Q$ with $H$ and $Q$ finitely generated and $H$ weakly convex, then $\operatorname{obdim}G\geq\operatorname{obdim}H+\operatorname{obdim}Q$.

$\textbf{Example:}$ Let $G=F_{n}^{n}$. Define $\phi:G\to\mathbb{Z}$ by sending the basis elements of each factor to $1\in\mathbb{Z}$. Let $H=\operatorname{Ker}(\phi)$. It is easy to see that $H$ contains a copy of $F_{n}^{n}$ and thus $\operatorname{obdim}(G)=\operatorname{obdim}(H)=2n$. Therefore $G=H\rtimes\mathbb{Z}$, but $\operatorname{obdim}(H)+\operatorname{obdim}(\mathbb{Z})=2n+1>\operatorname{obdim}(G)$. It follows that $H$ is not weakly convex, and one knows by e.g. Bestvina-Brady Morse theory that $H$ is of type $F_{n-1}$ (in particular, it is finitely generated for $n\geq 2$, finitely presented for $n\geq 3$, etc.). This shows that in corollary 7 weak convexity of $H$ is a necessary assumption.

By finding suitable obstruction complexes, and quoting heavily from the theory on structures of Lie groups and Lie algebras, Bestvina-Feighn prove

Theorem 8: Let $G$ be a connected semisimple Lie group, $K \subset G$ a maximal compact subgroup, $G/K$ the associated contractible manifold, and $\Gamma$ a lattice in $G$. If $\Gamma$ acts properly discontinuously on a contractible manifold $W$, then $\rm{dim}W \geq \rm{dim}G/K$.

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