Proof of Theorem E

In the ‘only if’ part, condition (i) follows from [Reference Bridson and HaefligerBH99, II.1A.6]. The proof of condition (ii) is identical to that in [Reference Ballmann and BuyaloBB96, Theorem 7.1], and the proof of condition (iii) was given in [Reference Norin, Osajda and PrzytyckiNOP22, Section 2]. For the proof of the ‘if’ part, suppose that a triangle complex X satisfies conditions (i)–(iii). By condition (i) and [Reference Bridson and HaefligerBH99, II.1A.6], we have that X is locally $\mathrm {CAT}(0)$ at any interior point of a triangle.

Consider now an edge e of X. Let $\mathrm {St}(e)$ be the union of all the closed triangles containing e. We will show that $\mathrm {St}(e)$ is $\mathrm {CAT}(0)$ , which implies that X is locally $\mathrm {CAT}(0)$ at any interior point x of e, since the metrics on $\mathrm {St}(e)$ and on X coincide on a sufficiently small neighbourhood of x. Let $Y\subset \mathrm {St}(e)$ be the union of the triangles T for which there exists a point on e with positive geodesic curvature in T. By condition (ii), there is at most one such triangle of given isometry type $T_0$ and given embedding $e\subset T_0$ , so Y has finitely many triangles. We denote this number by $m(e)$ for future reference. For each triangle T of $\mathrm {St}(e)$ outside Y, denote $Y_T=Y\cup T$ . By conditions (i) and (ii), and by [Reference Ballmann and BuyaloBB96, Theorem 7.1], we have that each $Y_T$ is locally $\mathrm {CAT}(0)$ , hence, $\mathrm {CAT}(0)$ by Theorem 2.1 (note that a geodesic in $Y_T$ might enter and exit a given triangle infinitely many times). Furthermore, the inclusion $Y\subset Y_T$ is an isometric embedding, since points of e have nonpositive geodesic curvature in T. By [Reference Bridson and HaefligerBH99, II.11.3], the union $\mathrm {St}(e)$ of $Y_T$ is $\mathrm {CAT}(0)$ , as desired.

Consider now a vertex v of X. After possibly subdividing X, we can assume that in each triangle T containing v, the local geodesic $\gamma _T$ starting at v and bisecting the angle of T at v ends at the opposite side of T. We will prove that the union $\mathrm {St}(v)$ of all the closed triangles and edges containing v is $\mathrm {CAT}(0)$ . This will imply that X is locally $\mathrm {CAT}(0)$ at v, since the metrics on $\mathrm {St}(v)$ and on X coincide on a sufficiently small neighbourhood of v.

To justify the Claim, we employ the idea of a taut string [Reference Bridson and HaefligerBH99, Chapter I.7.20]. Let $\theta $ be the minimum of $\pi $ and the minimum length of an edge in $\mathrm {lk}_v^X$ , and set $N=2+\frac {\pi }{\theta }$ . Let $M=1+ N(1+\max _e m(e))$ , where $m(e)$ is defined as above and the maximum is taken over all the edges e of $\mathrm {St}(v)$ containing v. For each such edge e, let $\mathrm {St}'(e)\subset \mathrm {St}(e)$ be the closure of the component containing e of $\mathrm {St}(e)\setminus \bigcup _{T\subset \mathrm {St}(e)}\gamma _T$ , for $\gamma _T$ as above. Since $\mathrm {St}'(e)$ is convex in $\mathrm {St}(e)$ , it is $\mathrm {CAT}(0)$ [Reference Bridson and HaefligerBH99, II.1.15(1)].

For $x,y\in \mathrm {St}(v)$ , a string between x and y is a sequence of edges $e_1,\ldots , e_n$ of $\mathrm {St}(v)$ containing v and points $x_0=x,x_1,\ldots , x_n=y$ with $\mathrm {St}'(e_i)$ containing both $x_{i-1}$ and $x_i$ . The length of the string is the sum $\Sigma _{i=1}^nd_i(x_{i-1},x_i)$ , where $d_i$ is the metric on $\mathrm {St}'(e_i)$ . The distance between x and y in $\mathrm {St}(v)$ is the infimum of the lengths of strings between x and y. A string is taut, if $n\leq 2$ , or

• • for each $0\leq i\leq n,$ the point $x_i$ is distinct from v, and

• • for each $0<i<n,$ the point $x_i$ belongs to one of the $\gamma _T$ defined above and

• • for each $0<i<n,$ the concatenation of geodesics $x_{i-1}x_i$ in $\mathrm {St}'(e_i)$ and $x_{i}x_{i+1}$ in $\mathrm {St}'(e_{i+1})$ is a geodesic in $\mathrm {St}'(e_i)\cup \mathrm {St}'(e_{i+1})$ .

We now justify that for each string $(e_i),(x_i)$ between x and y, we can find a taut string between x and y whose length does not exceed the length of $(e_i),(x_i)$ . Indeed, by discarding some $x_i$ , we can first assume that consecutive $e_i$ are distinct, and so for each $0<i<n$ , the point $x_i$ belongs to one of the $\gamma _T$ . Since $\gamma _T$ are compact, there is a choice of $x^{\prime }_i$ in the same $\gamma _T$ as $x_i$ , minimising the length of the string $(e_i),(x^{\prime }_i)$ . Then the concatenation of geodesics $x^{\prime }_{i-1}x^{\prime }_i$ in $\mathrm {St}'(e_i)$ and $x^{\prime }_{i}x^{\prime }_{i+1}$ in $\mathrm {St}'(e_{i+1})$ is a geodesic in $\mathrm {St}'(e_i)\cup \mathrm {St}'(e_{i+1})$ . Finally, if an $x^{\prime }_{i-1}$ equals v, and $x^{\prime }_{i}\neq x_n$ , then for $x^{\prime }_i\in \gamma _T$ , the subpath of $\gamma _T$ from $x^{\prime }_{i-1}$ to $x^{\prime }_{i}$ is a geodesic in both $\mathrm {St}'(e_i)$ and $\mathrm {St}'(e_{i+1})$ , and consequently, we can discard $x^{\prime }_{i}$ and $e_i$ from the string. Repeating the argument, we arrive at $i-1=n-1$ or $i-1=n$ . Analogously, we obtain $i-1=1$ or $i-1=0$ , and so $n\leq 2$ .

We will now show that a taut string satisfies $n\leq N$ . We can assume $n>2$ , and so none of $x_i$ equals v. For $0<i<n,$ let $\theta _i$ be the Alexandrov angle at $x_{i}$ in $\mathrm {St}'(e_{i+1})$ between the geodesics $x_{i}x_{i+1}$ and $x_{i}v$ . The concatenation of geodesics $x_{i-1}x_i$ in $\mathrm {St}'(e_i)$ and $x_{i}x_{i+1}$ in $\mathrm {St}'(e_{i+1})$ is a geodesic in $\mathrm {St}'(e_i)\cup \mathrm {St}'(e_{i+1})$ , which is $\mathrm {CAT}(0)$ by [Reference Bridson and HaefligerBH99, II.11.3]. Consequently, by the definition of $\theta $ , we have $\theta _i\geq \theta _{i-1}+\theta $ . Thus, $\pi \geq \theta _{n-1}\geq (n-2)\theta +\theta _1$ , and so, $n-2\leq \frac {\pi }{\theta }=N-2$ .

Since there are finitely many isometry types of simplices in $\mathrm {St}(v)$ , and each taut string satisfies $n\leq N$ , the distance between x and y in $\mathrm {St}(v)$ is realised by the length of some taut string $(e_i),(x_i)$ . Then the concatenation of all the geodesics $x_{i-1}x_i$ in $\mathrm {St}'(e_i)$ is a geodesic between x and y, proving that $\mathrm {St}(v)$ is geodesic. Furthermore, for any geodesic $\gamma $ from x to y in $\mathrm {St}(v)$ , one can choose points on $\gamma $ forming a taut string. Since any taut string satisfies $n\leq N$ , and we have that $\gamma $ intersects the interiors of at most $1+n(1+\max _e m(e))$ triangles, the Claim follows.

Returning to the proof of Theorem E, we follow the scheme in [Reference Ballmann and BuyaloBB96, Theorem 7.1] to find a sequence of $\mathrm {CAT}(0)$ spaces Gromov–Hausdorff converging to $\mathrm {St}(v)$ . Namely, realise each (isometry type of a) triangle T of $\mathrm {St}(v)$ as $T\subset \mathbb {R}^2$ , with metric induced from some smooth Riemannian metric of nonpositive Gaussian curvature on $\mathbb {R}^2$ defined in a neighbourhood U of T. Denote by $e,f$ the edges of T containing v and by g the remaining edge of T. Let l denote the length of e. For each $n>0$ , we decompose e into paths $a^1\cdot a^2 \cdots a^n$ of length $\frac {l}{n}$ , and we define $\kappa ^k$ to be the integral of the geodesic curvature along $a^k$ . Let $e_n$ be the piecewise geodesic in U that starts at v tangent to e, has n locally geodesic pieces of length $\frac {l}{n}$ and exterior angle at the k-th breakpoint that equals $\kappa ^k$ . For n sufficiently large the path $e_n$ exists, and they $C^1$ -converge to e as n tends to $\infty $ . We define paths $f_n$ analogously. We define $g_n$ to be any piecewise geodesics joining the endpoints of $e_n$ and $f_n$ and $C^1$ -converging to g. This gives us a triangle $T_n\subset U$ bounded by $e_n,f_n$ and $g_n$ , whose boundary is piecewise geodesic (one can pass to a union of triangles with locally geodesic boundary by subdividing). Furthermore, we have a map $T_n\to T$ , whose restriction to $e_n,f_n$ preserves length and which is bilipschitz with the bilipschitz constant converging to $1$ as n tends to $\infty $ . Glueing various $T_n$ along the sides corresponding to the ones of T that we glued to form $\mathrm {St}(v)$ yields a triangle complex that we call $\mathrm {St}(v)_n$ . Then $\mathrm {St}(v)$ is a Gromov–Hausdorff limit of $\mathrm {St}(v)_n$ . Note that since $\mathrm {St}(v)$ satisfied conditions (i)–(iii), we have that $\mathrm {St}(v)_n$ satisfies conditions (i)–(iii). Since $\mathrm {St}(v)_n$ has locally geodesic edges, and is geodesic by the Claim (applied to $\mathrm {St}(v)_n$ instead of $\mathrm {St}(v)$ ), the same proof as for [Reference Ballmann and BuyaloBB96, Theorem 7.1, case 1] shows that $\mathrm {St}(v)_n$ is locally $\mathrm {CAT}(0)$ , hence, $\mathrm {CAT}(0)$ by Theorem 2.1. Consequently, by [Reference Bridson and HaefligerBH99, II.3.10], we have that $\mathrm {St}(v)$ is $\mathrm {CAT}(0)$ .

Proof of Theorem A

By passing to a subdivision (see [Reference Norin, Osajda and PrzytyckiNOP22, Lemma 2.1]), we can assume that G acts without weak inversions. By Proposition 3.1, we can assume that X contains a thick G-c.s. $Z_1$ . We will prove that G contains a nonabelian free group. By passing to a connected component of $Z_1$ and its stabiliser $G'$ in G (which is finitely generated, since it acts properly and cocompactly on a connected complex), we can assume that $Z_1$ is connected. If $Z_1$ contains a closed edge-path that is not contractible in $Z_1$ , repeatedly attaching to $Z_1$ the images of reduced disc diagrams and their G-translates, we obtain an increasing sequence $Z_1\subset Z_2\subset \cdots $ of connected essential G-c.s., such that their union $X'$ is simply connected. By Corollary 2.3, we have that $X'$ is locally $\mathrm {CAT}(0)$ , and so $X'$ is $\mathrm {CAT}(0)$ by Theorem 2.1. It remains to apply Proposition 3.2.

Proof. Since W has finitely many isometry types of simplices, by Remark 2.4, all elements of A act as loxodromic isometries on W. By [Reference Bridson and HaefligerBH99, II.7.20(1)], we have $\mathrm {Min}(A) = Y \times \mathbb {R}^n$ with A preserving the product structure and acting trivially on Y. By [Reference Bridson and HaefligerBH99, II.7.20(2)], we have $n\leq 2$ , but since A acts freely by simplicial isometries, we have $n=2$ , and so Y is a point, as desired.

Proof of Lemma 4.1

Let e be an edge of Z of degree $\geq 3$ , and let x be a point in the interior of e. Let $b_1,b_2,b_3,$ be geodesics starting at x perpendicularly to e contained in distinct triangles $T_1,T_2,T_3$ . For each $i=1,2,3,$ the set of starting directions at points in $\partial T_i$ of geodesics intersecting $b_i$ at angle $<\frac {\pi }{6}$ has positive Liouville measure (see [Reference Ballmann and BrinBB95, Section 3]). Let S denote the union of all the open edges in the links $\mathrm {lk}_y^Z$ for all $y\in Z^1\setminus Z^0$ , with the (infinite) Liouville measure. We say that $(\xi _j)_j\in S^{\mathbb {Z}}$ with $\xi _j\in \mathrm {lk}_{y_j}^Z$ determines a locally geodesic oriented line $\gamma $ in $Z\setminus Z^0$ transverse to $Z^1$ if $\gamma $ intersects $Z^1$ exactly at points $y_j$ in directions $\xi _j$ , in that order. Since G acts on Z properly and cocompactly, the set of $(\xi _j)_j\in S^{\mathbb {Z}}$ that determine locally geodesic oriented lines projects to a full measure subset in each coordinate S (see [Reference Ballmann and BrinBB95, Section 3], which relies on [Reference Cornfeld, Fomin and SinaĭCFS82, Chapter 6]). Consequently, for $i=1,2,3,$ there exists a locally geodesic ray $\gamma _i$ in Z starting at an interior point $x_i$ of $b_i$ at angle $<\frac {\pi }{6}$ from $b_i$ , disjoint from $Z^0$ and transverse to $Z^1$ . Let $a_i=xx_i\subset b_i$ . See Figure 1.

Figure 1

The geodesic ray $\gamma _1$ .

Since X is $\mathrm {CAT}(0)$ , by Theorem 2.2, we have that $\gamma _i$ are geodesic rays in X and $a^{-1}_i\cdot a_j$ are geodesics in X. Since each $\gamma ^{-1}_i\cdot a^{-1}_i\cdot a_j\cdot \gamma _j$ is a piecewise geodesic with angles $>\frac {5\pi }{6}$ at the two breakpoints, by [Reference Bridson and HaefligerBH99, II.9.3], the rays $\gamma _i,\gamma _j$ are not asymptotic and they determine points at distance $>\frac {2\pi }{3}$ in the Tits boundary of X. In particular, Z cannot be quasi-isometric to $\mathbb {R}$ since it contains three pairwise nonasymptotic geodesic rays. This proves (i).

For (ii), assume for contradiction, that G is virtually $\mathbb {Z}^2$ generated by elements $g,h$ . Let $\alpha , \beta $ be edge-paths in Z connecting a basepoint $y\in Z^0$ to $gy,hy$ , respectively. Then the concatenation $\alpha \cdot g\beta \cdot h\alpha ^{-1}\cdot \beta ^{-1}$ is a closed edge-path, and since X is simply connected, there is a reduced disc diagram $D\to X$ with that boundary path. Let $Z'\subset X$ be the connected thick G-c.s. obtained from Z by adding the translates under G of the image of D. The complex $Z'$ is locally $\mathrm {CAT}(0)$ by Corollary 2.3. Let $\widetilde Z'\to Z'$ be the universal cover of $Z'$ , which is $\mathrm {CAT}(0)$ by Theorem 2.1. The action of G on $Z'$ lifts to an almost free action of a group $\widetilde G$ on $\widetilde Z'$ fitting into the short exact sequence $\pi _1 Z'\to \widetilde G\to G$ . Since $D\to Z'$ lifts to $D\to \widetilde Z'$ , we have commuting $\tilde g, \tilde h\in \widetilde G$ mapping to $g,h\in G$ , and, hence, generating a subgroup $A< \widetilde G$ isomorphic to $\mathbb {Z}^2$ .

Since A acts almost freely and is torsion free, we have that it acts freely on $\widetilde Z'$ . By Lemma 4.2 applied with $W=\widetilde Z'$ , there is an isometrically embedded A-cocompact subcomplex $E\subset \widetilde Z'$ isometric to the Euclidean plane. We now justify that the composition $\phi \colon E\subset \widetilde Z'\to Z'\subset X$ is an isometric embedding.

Indeed, for two triangles $T,T'$ of E containing a common edge $e'$ , the sum of the geodesic curvatures in $\phi (T)$ and $\phi (T')$ at any point of $\phi (e')$ equals $0$ , and so the geodesic curvature at $\phi (e')$ in any triangle of X distinct from $\phi (T),\phi (T')$ is nonpositive. Consequently, $\phi $ is a local isometric embedding at $e'$ . Furthermore, since E is isometric to the Euclidean plane, for any geodesic $\gamma $ in E passing through a vertex v, the angle (see Section 2) between the incoming and outgoing directions of $\gamma $ at v equals $\pi $ . Since the map that $\phi $ induces between $\mathrm {lk}_v^E$ and $\mathrm {lk}_{\phi (v)}^X$ is locally injective, the angle between the incoming and outgoing directions of $\phi (\gamma )$ at $\phi (v)$ equals $\pi $ as well. By Theorem 2.2, we have that $\phi (\gamma )$ is a geodesic. Thus, $\phi $ is an isometric embedding, as desired.

Since the image of A in G is of finite index, it acts cocompactly on $Z'$ . Consequently, the geodesic rays $\gamma _i$ from the proof of part (i) are at bounded distance from $\phi (E)$ in $Z'$ . Since X is $\mathrm {CAT}(0)$ , we obtain that each $\gamma _i$ is asymptotic to a geodesic ray in $\phi (E)$ and these three rays are pairwise at angle $>\frac {2\pi }{3}$ in $\phi (E)$ , which is a contradiction.

Proof. For part (i), note that if for some indices $\lambda ,\mu $ , we have $v^{\lambda }=v^{\mu }$ , then $w^{\lambda }=w^{\mu }$ , since the point $y^{\lambda }$ in $\mathrm {lk}_{v^{\lambda }}^X$ does not depend on $\Gamma ^{\lambda }_1$ . Consequently, distinct edges $v^{\lambda } w^{\lambda }$ might intersect only along $w^{\lambda }$ . Thus, their union $V\subset X$ is a forest, and so the quotient map $q\colon X\to X^*$ collapsing each component of V into a point is a homotopy equivalence. Similarly, the subcomplex $V_{\mathcal F}=p_{\mathcal F}^{-1}(V)\subset X_{\mathcal F}$ is a forest, and so the quotient map $q_{\mathcal F}\colon X_{\mathcal F}\to X^*$ collapsing each component of $V_{\mathcal F} $ into a point is also a homotopy equivalence. Thus, the identity $q_{\mathcal F} =q\circ p_{\mathcal F}$ implies part (i).

Part (ii) follows from the fact that the maps that $p_{\mathcal F}$ induces between the links of $X_{\mathcal F}$ and X are locally injective and from Theorem E.

The map $p_{\mathcal F}$ is a local isometry at the open edges outside $V_{\mathcal F}$ . Thus, for part (iii), it suffices to justify that the links of the vertices in each $p_{\mathcal F}^{-1}(v^{\lambda })$ have no leaves. But each such link is a cycle $\Gamma ^{\mu }_1$ (for $v^{\mu }=v^{\lambda }$ ) or a clover, as desired.

Proof. We fix an increasing sequence $Z_k\subset X$ of connected essential G-c.s. exhausting X. A multifolding $(X',(Z^{\prime }_k),p')$ is a:

1. (i) $\mathrm {CAT}(0)$ triangle complex $X'$ with an action of G,

2. (ii) a sequence $(Z^{\prime }_k)$ of essential G-c.s. exhausting $X'$ and

3. (iii) a G-equivariant simplicial map $p'\colon X'\to X$ that

   • • maps bijectively the set of triangles of each $Z^{\prime }_k$ to the set of triangles of $Z_k$ and

   • • whose restriction $Z^{\prime }_k\to Z_k$ is a homotopy equivalence.

We introduce a partial order $\leq $ on the set of multifoldings, writing $(X',(Z^{\prime }_k),p')\leq (X",(Z^{\prime \prime }_k),p")$ (or, shortly, $X'\leq X"$ ) if there is a G-equivariant simplicial map $r\colon X"\to X'$ satisfying $p"=p'\circ r$ . Multifoldings $X',X"$ are equivalent if $X'\leq X"$ and $X"\leq X'$ . Let $\mathcal X$ be the set of equivalence classes of multifoldings.

We claim that every chain of elements $X^{\prime \lambda }$ in $\mathcal X$ (where $\lambda $ is an index) has an upper bound. Indeed, denote by $p^{\prime }_k$ the restriction of $p'$ to $Z^{\prime }_k$ and write $(Z^{\prime }_k,p^{\prime }_k)\leq (Z^{\prime \prime }_k,p^{\prime \prime }_k)$ whenever there is a G-equivariant simplicial map $r\colon Z^{\prime \prime }_k\to Z^{\prime }_k$ satisfying $p^{\prime \prime }_k=p^{\prime }_k\circ r$ . For each k, since G acts properly and cocompactly on $Z^{\prime \lambda }_k$ , by the first bullet point, we have that $(Z^{\prime \lambda }_k, p^{\prime \lambda }_k)$ can take on only finitely values up to the appropriate equivalence. Thus, there exists a largest element among the $(Z^{\prime \lambda }_k, p^{\prime \lambda }_k)$ , which we call $Z^{\prime \infty }_k$ . Furthermore, since $(Z^{\prime }_{k+1},p^{\prime }_{k+1})\leq (Z^{\prime \prime }_{k+1},p^{\prime \prime }_{k+1})$ implies $(Z^{\prime }_{k},p^{\prime }_k)\leq (Z^{\prime \prime }_{k},p^{\prime \prime }_k)$ , we have natural injective maps $Z^{\prime \infty }_k\to Z^{\prime \infty }_{k+1}$ . Let $X^{\prime \infty }$ be their direct limit, equipped with the limit map $p^{\prime \infty }$ to X. Since each $Z^{\prime \infty }_k$ is locally $\mathrm {CAT}(0)$ (Corollary 2.3), we have that $X^{\prime \infty }$ is locally $\mathrm {CAT}(0)$ (Theorem E).

To prove that $X^{\prime \infty }$ is an upper bound for our chain in $\mathcal X$ , by Theorem 2.1, it remains to prove that $X^{\prime \infty }$ is simply connected. Let $\alpha $ be a closed edge-path in the $1$ -skeleton of $X^{\prime \infty }$ , and fix k, such that $\alpha $ lies in $Z^{\prime \infty }_k$ . Fix $\lambda $ with $Z^{\prime \lambda }_k=Z^{\prime \infty }_k$ , and keep the notation $\alpha $ for its copy in $Z^{\prime \lambda }_k$ . Since $X^{\prime \lambda }$ is simply connected, there is a disc diagram $D\to X^{\prime \lambda }$ with boundary path $\alpha $ . Fix l, such that the image of D is contained in $Z^{\prime \lambda }_l$ . Since, by the second bullet point, the induced map $\pi _1Z^{\prime \infty }_l\to \pi _1Z^{\prime \lambda }_l$ is an isomorphism, we have that $\alpha $ is trivial in $\pi _1Z^{\prime \infty }_l$ and, hence, in $\pi _1X^{\prime \infty }$ . Consequently, by the Kuratowski–Zorn lemma, there is a maximal element $X'\in \mathcal X$ .

We now prove that none of the links of $X'$ are unfoldable. Otherwise, suppose that $X'$ has a vertex v whose link $\Gamma =\Gamma _1\vee \Gamma _2$ is unfoldable at a point corresponding to an edge $vw$ of $X'$ . Let $\mathcal F=\{g\Gamma _1\}$ , for $g\in G$ . Since G acts without weak inversions, we have $gv\neq hw$ , for all $g,h\in G$ . Thus, we can define the folding over $\mathcal F$ , which we denote by $X_{\mathcal F}\to X'$ . The action of G on $X'$ lifts to an action of G on $X_{\mathcal F}$ . For each essential G-c.s. $Z'\subset X'$ , let $Z_{\mathcal F}\subset X_{\mathcal F}$ be the closure of the union of all the open triangles of $X_{\mathcal F}$ mapping into $Z'$ . Note that if $Z'$ does not contain v, or if $Z'$ contains v, but $\mathrm {lk}^{Z'}_v$ is contained in $\Gamma _1$ or $\Gamma _2$ (in the latter case, $\mathrm {lk}^{Z'}_v\subset \bigcap _{g\in \mathrm {Stab}(v)}g\Gamma _2$ ), then $Z_{\mathcal F}\to Z'$ is an isometry. Otherwise, $\mathrm {lk}^{Z'}_v$ contains edges lying on both the cycle $\Gamma _1$ and the clover $\Gamma _2$ . Since $Z'$ is essential, the link $\mathrm {lk}^{Z'}_v$ is the wedge of $\Gamma _1$ and a clover, so it is unfoldable. Then $Z_{\mathcal F}\to Z'$ is the folding over the family $\{g\Gamma _1\}$ , for $g\in G$ . By Lemma 5.2(i,iii), we have that $Z_{\mathcal F}$ is essential and $Z_{\mathcal F}\to Z'$ is a homotopy equivalence. By Lemma 5.2(i,ii), we have that $X_{\mathcal F}$ is simply connected and locally $\mathrm {CAT}(0)$ , and, hence, $\mathrm {CAT}(0)$ by Theorem 2.1. Consequently, we have $X_{\mathcal F}\in \mathcal X$ and $X_{\mathcal F}>X'$ , contradicting the maximality of $X'$ . Thus, none of the links of $X'$ are unfoldable.

For the last assertion, note that, by Lemma 4.1 applied to X, we have that G is neither virtually cyclic, nor virtually $\mathbb {Z}^2$ . Moreover, G is finitely generated, since it acts properly and cocompactly on $Z_1$ , which is connected. Consequently, if $X'$ does not have edges of degree $3$ , then by Proposition 3.1, we have that G contains a nonabelian free group, as desired.

Proof. Suppose that $\gamma $ passes through a vertex v with incoming and outgoing directions at angle $>\pi +\kappa ,$ for some $\frac {\pi }{2}>\kappa >0$ . Let R be the translation length of g. We will prove that $M=R\lceil \frac {2\pi }{\kappa }\rceil $ satisfies the lemma. Otherwise, let $x,y$ be points in a closed metric ball disjoint from $\gamma $ , such that the projections $x',y'$ of $x,y$ to $\gamma $ are at distance $>M$ .

There are at least $n=\lceil \frac {2\pi }{\kappa }\rceil $ translates of v under $\langle g \rangle $ on $x'y'$ distinct from $x',y'$ . We denote these translates by $v^{\prime }_1,\ldots , v^{\prime }_n$ , in the order in which they appear on $x'y'$ . By the continuity of the projection map, there are points $v_1,\ldots ,v_n$ lying on the geodesic $xy$ in that order, such that each $v_i'$ is the projection of $v_i$ to $\gamma $ . We additionally denote $v_0=x, v_0'=x', v_{n+1}=y,v^{\prime }_{n+1}=y'$ . For $0\leq i\leq n$ , let $\beta _i$ denote the geodesic quadrilateral $v_iv_{i+1}v^{\prime }_{i+1}v^{\prime }_iv_i$ . The sum of the four Alexandrov angles of each $\beta _i$ is $\leq 2\pi $ [Reference Bridson and HaefligerBH99, II.2.11], so the sum of all the Alexandrov angles of all $\beta _i$ is $\leq (n+1)2\pi $ .

On the other hand, for $0\leq i<n$ , the sum of the Alexandrov angles of $\beta _i$ and $\beta _{i+1}$ at $v_{i+1}$ is $\geq \pi $ . We will now prove that the sum of the Alexandrov angles of $\beta _i$ and $\beta _{i+1}$ at $v^{\prime }_{i+1}$ is $>\pi +\kappa $ . Indeed, if one of them is not equal to the angle in the usual sense (see Section 2), then it equals $\pi $ . However, since $v^{\prime }_{i+1}$ is the projection of $v_{i+1}$ , the second Alexandrov angle is $\geq \frac {\pi }{2}$ , so their sum is $\geq \frac {3\pi }{2}$ , as desired. Consequently, we have $n(\pi +\pi +\kappa )< (n+1)2\pi $ , and so $n\kappa < 2\pi $ , which is a contradiction.

Proof. Consider the space of geodesic rays $\rho \colon [0,\infty )\to X$ representing $\xi $ , with the pseudometric $d(\rho _1,\rho _2)=\inf _{t_1,t_2}d(\rho _1(t_1),\rho _2(t_2))$ . Identifying $\rho _1$ with $\rho _2$ for $d(\rho _1,\rho _2)=0,$ we obtain a metric space whose metric completion $X_{\xi }$ is $\mathrm {CAT}(0)$ [Reference LeebLee00, Proposition 2.8]. Since X has geometric dimension $\leq 2$ (see [Reference KleinerKle99]), by [Reference CapraceCap09, Remark after Corollary 4.4], we have that $X_{\xi }$ has geometric dimension $\leq 1$ (more precisely, as we learned from Pierre-Emmanuel Caprace, for any $\rho $ representing $\xi $ , the space $X_{\xi }\times \mathbb {R}$ embeds isometrically in the pointed ultralimit of $(X,\rho (n))_n$ , which has geometric dimension $\leq 2$ by [Reference LytchakLyt05, Lemma 11.1]). Since G fixes $\xi $ , the action of G on X induces an action of G on $X_{\xi }$ .

We will now justify that a complete $\mathrm {CAT}(0)$ space $X_{\xi }$ of geometric dimension $\leq 1$ is an $\mathbb {R}$ -tree, which we learned also from Pierre-Emmanuel Caprace. For a geodesic triangle $xyz$ in $X_{\xi }$ , let $x'$ be the projection of x to the geodesic $yz$ . If $x'\neq x,y$ , then the direction of the geodesic $x'x$ in $\mathrm {lk}_{x'}^{X_{\xi }}$ is distinct from that of the geodesic $x'y$ . Thus, the geodesic $xy$ must pass through $x'$ , since otherwise mapping it to $\mathrm {lk}_{x'}^{X_{\xi }}$ would give a path between distinct points of a discrete set. Analogously, the geodesic $xz$ passes through $x'$ , and so $xyz$ is $0$ -thin, justifying that $X_{\xi }$ is an $\mathbb {R}$ -tree.

Suppose, first, that there is $h\in G$ acting loxodromically on $X_{\xi }$ . Let M be the constant given by Lemma 6.2 for $\gamma $ . Let $\rho $ be a ray in $\gamma $ representing $\xi $ . Since h acts loxodromically on $X_{\xi }$ , after possibly replacing h by its power, we can assume $d(\rho ,h\rho )>M$ . Assume, without loss of generality, that the Busemann function (see [Reference Bridson and HaefligerBH99, II.8.17 and II.8.20]) satisfies $B_{\xi }(\rho (0))\leq B_{\xi }(h\rho (0))$ . Then for each $t\geq 0$ , the projection $p(t)$ of $\rho (t)$ to $h\gamma $ is contained in $h\rho $ , and for $D=d(\rho (0),h\rho (0))$ , we have $d(\rho (t),p(t))\leq D$ . Pick $k\in \mathbb {N}$ with $kM>2D$ . Then, by the triangle inequality, we have $d(p(0),p(2kM))\geq 2kM-2D>kM$ . Consequently, there is $0\leq n<k$ , such that $d(p(2nM),p(2(n+1)M))>M$ . Thus, the closed ball of radius M centred at $\rho ((2n+1)M)$ is disjoint from $h\gamma $ and contains points $\rho (2nM),\rho (2(n+1)M)$ whose projections to $h\gamma $ are at distance $>M$ . This contradicts the choice of M.

Consequently, G has a global fixed point in $X_{\xi }$ (which might not be represented by a geodesic ray but be a point added in the completion). Thus, there is $D>0$ , such that for any $\varepsilon>0$ , there is a geodesic ray $\rho '$ representing $\xi $ at distance $\leq D$ from $\rho $ and satisfying $d(\rho ',g\rho ')<\varepsilon $ for each $g\in G$ . Consider the homomorphism $\psi \colon G\to \mathbb {R}$ defined by $\psi (g)=B_{\xi }(gx)-B_{\xi }(x)$ for any $x\in X$ . We will now justify that $\psi $ has discrete image. Otherwise, for any $\varepsilon>0$ and $\rho '$ as above, there is $t>0$ , such that $d(\rho '(t),g\rho '(t))<2\varepsilon $ , but g does not fix a point of X. This contradicts Remark 2.5 applied with Y containing the D-neighbourhood of $\gamma $ .

Let K be the kernel of $\psi $ . Arguing as in the previous paragraph, we obtain that every $g\in K$ fixes a point of X. By [Reference Norin, Osajda and PrzytyckiNOP22, Theorem 1.1(i)], every finitely generated subgroup of K fixes a point of X. Since K acts almost freely, we have that K is finite and so G is virtually cyclic.

Proof. By Lemma 6.2, g is rank $1$ in the sense of [Reference Bestvina and FujiwaraBF09, Definition 5.1]. By Lemma 6.3, we can assume that G does not have a finite index subgroup fixing a limit point of $\gamma $ . Then there is $f\in G$ with $\gamma $ and $f\gamma $ having disjoint limit point pairs. Consequently, by [Reference Bestvina and FujiwaraBF09, Proposition 5.9], for some n, the elements $g^n$ and $fg^nf^{-1}$ generate a nonabelian free group.

Proof. Let $Z\subset X$ be an essential G-c.s. containing v, such that $\mathrm {lk}^{Z}_v$ contains $\xi _i,\eta _i$ , with $d^{Z}_v(\xi _i,\eta _i)=\pi ,$ for $i=1,\ldots , n$ . By [Reference Norin, Osajda and PrzytyckiNOP22, Lemma 5.4] (which was stated in terms of the compact quotient but has the same proof for proper and cocompact actions), for any $\varepsilon>0$ , there is a path $\omega =\omega _1\cdots \omega _{3n}$ in Z, such that

• • paths $\omega _j$ are local geodesics in Z, and

• • there are $g_0=\mathrm {id}, g_1,\ldots , g_n=g\in G$ such that paths $\omega _{3i+1}$ start at $g_iv$ , paths $\omega _{3i}$ end at $g_{i}v$ , and except for that $\omega $ is disjoint from the vertex set $Z^0$ and transverse to $Z^1$ , and

• • the starting direction of $\omega _{3i+1}$ is at distance $<\frac {\varepsilon }{2}$ to $g_i\xi _{i+1}$ in $\mathrm {lk}^{Z}_{g_iv}$ , and the ending direction of $\omega _{3i}$ is at distance $<\frac {\varepsilon }{2}$ to $g_i\eta _i$ and

• • at the remaining breakpoints, $\omega _{j}$ and $\omega _{j+1}$ are at angle $>\pi -\varepsilon $ (and $\leq \pi $ since outside $Z^0$ ).

Since $\omega _j$ are disjoint from $Z^0$ and transverse to $Z^1$ , by Theorem 2.2, they are geodesics in X. The last two bullet points hold in X as well. In particular, at all the breakpoints $\omega _{j}$ and $\omega _{j+1}$ are at angle $>\pi -\varepsilon $ . By [Reference Ballmann and BrinBB95, Lemma 2.5], the geodesic $\gamma $ in X with the same endpoints as $\omega $ starts and ends in directions at distance $<(3n-1)\varepsilon $ to $\xi _1,g\eta _n$ . Consequently, for $\varepsilon $ sufficiently small, by Theorem 2.2, we have that $\bigcup _{l\in \mathbb {Z}}g^l\gamma $ is a curved axis for g.

Proof. To prove that G is virtually cyclic or contains a nonabelian free group in each case we will show the existence of a curved axis in X (or in a different $\mathrm {CAT}(0)$ triangle complex $\overline X$ ) for an element of G, since then the proposition follows from Proposition 6.4.

Assume first that X is not G-equivariantly isometric to a piecewise Euclidean triangle complex $X'$ . Then by [Reference Ballmann and BrinBB95, Proposition 2.11] there is

1. (i) a point in the interior of a triangle of X with negative Gaussian curvature, or

2. (ii) a point in the interior of an edge of X with negative sum of geodesic curvatures of some two incident triangles, or

3. (iii) a vertex v of X with $\mathrm {lk}^X_v$ a circle of length $>2\pi $ .

In case (iii), or, more generally, if $\mathrm {lk}^X_v$ has a component C that is a circle of length $>2\pi $ , let $\xi _1,\eta _1$ be points at distance $\pi $ in C, and let $\eta _2,\xi _2$ be their antipodal points. Applying Lemma 7.3 with $n=2$ , we obtain a curved axis. In cases (i) and (ii), by [Reference Norin, Osajda and PrzytyckiNOP22, Lemma 5.5], there is a $\mathrm {CAT}(0)$ triangle complex $\overline X$ , obtained from X by a G-equivariant subdivision and a G-equivariant replacement of the smooth Riemannian metrics, with a vertex $u\in \overline X$ whose $\mathrm {lk}^{\overline X}_u$ is either

• • a circle of length $>2\pi $ or

• • a graph obtained from a family of disjoint circles $C_1,C_2,\ldots $ of length $2\pi $ by glueing them along a nontrivial arc b of length $<\pi $ .

The first bullet point brings us to case (iii). In the case of the second bullet point, let $\xi _1,\xi _2\in C_1\setminus b$ and $\eta _1,\eta _2\in C_2\setminus b$ be points at distance $\frac {\pi }{2}$ from the endpoints of b, with $d^{\overline X}_u(\xi _1,\eta _1)=d^{\overline X}_u(\xi _2,\eta _2)=\pi $ . Applying Lemma 7.3 with $n=2$ , we obtain a curved axis in $\overline X$ , as desired.

Thus, without loss of generality, we can assume that X is a piecewise Euclidean triangle complex. If X is not rational, then by [Reference Ballmann and BrinBB95, Proposition 7.7], applied to an essential G-c.s., there is a closed locally injective edge-path $\beta $ in some $\mathrm {lk}^X_v$ whose length is not commensurable with $\pi $ . In particular, by [Reference Ballmann and BrinBB95, Lemma 6.1(iii)], there are points $\xi ,\eta $ in $\mathrm {lk}^X_v$ at distance $>\pi +\delta ,$ for some $\delta>0$ (one could apply [Reference Norin, Osajda and PrzytyckiNOP22, Corollary 1.7] to find such $\xi ,\eta $ in $\beta $ , but it does not simplify the argument). Let $\beta _-$ (respectively, $\beta _+$ ) be the shortest path from $\xi $ (respectively, $\eta $ ) to $\beta $ . Since the length of $\beta $ is not commensurable with $\pi $ , there is a path $\beta _-\beta _0\beta _+$ with $\beta _0$ factoring through the universal cover of $\beta $ whose length equals $(2n-1)\pi +\delta '$ for some $n\in \mathbb {N}$ and $0\leq \delta '\leq \delta $ . Choosing $\xi _1=\xi ,\eta _1,\xi _2,\ldots , \eta _{n}$ as consecutive points at distance $\pi $ along that path, we have $d^{X}_v(\xi _1,\eta _n)>\pi $ . Applying Lemma 7.3, we obtain a curved axis.

Finally, if X is not extrational, let P be a patch of X with nontrivial $\psi =\psi (P)$ . Since P is planar, there is an element in $H_1(\overline P,\partial P)$ represented by a piecewise affine path $\alpha $ in $\overline P$ with endpoints in $\partial P$ and $\psi (\alpha )\neq 0$ . Let $\alpha $ be shortest among such paths, which exists since $\mathrm {Stab}(P)$ acts cocompactly on $\overline P$ . Note that then $\alpha $ does not intersect $\partial P$ except at its endpoints, since, otherwise, we could decompose it into two shorter paths, with $\psi $ nontrivial on at least one of them. Thus, the image of $\alpha $ in X (for which we keep the same notation) is a local geodesic in X that intersects the branching locus E exactly at its endpoints $x,x'$ . Let e (respectively, $e'$ ) be the segment of $\mathrm {lk}^X_x$ (respectively, $\mathrm {lk}^X_{x'}$ ) containing the point corresponding to the direction of $\alpha $ . By the shortness condition, we have that e (respectively, $e'$ ) has endpoints at distance $\geq \frac {\pi }{2}$ from x (respectively, $x'$ ), and so is of length $\geq \pi $ . They cannot both have length $\pi $ , since then we would have $\psi (\alpha )=0$ , so assume, without loss of generality, that the length l of e is $>\pi $ .

If $l> 2\pi $ , then it is easy to find points $\eta _1,\xi _1,\xi _2,\eta _2$ lying on l in that order and satisfying the hypothesis of Lemma 7.3 with $n=2$ . If $2\pi> l>\pi $ or $l=2\pi $ and the endpoints of e are distinct, then the construction of such points is given in the proof of [Reference Ballmann and BrinBB95, Lemma 7.6]. It remains to consider the case where $l=2\pi $ and where both endpoints of e are equal to a vertex y. Let $\Gamma _2$ be the graph obtained from $\mathrm {lk}^X_x$ by removing e. If $\Gamma _2$ contains a point z at distance $>\pi $ from y, then it is easy to find points $\xi _1,\eta _1,\xi _2,\eta _2$ on a geodesic from z to the midpoint of e satisfying the hypothesis of Lemma 7.3 with $n=2$ . Otherwise, $\Gamma _2$ is a clover, contradicting the assumption that $\mathrm {lk}^X_x$ is not unfoldable.

Proof. We have $z\neq x$ since edges are geodesics, and so, in particular, x does not lie in e. If for some z in the interior of e the geodesic $xz$ does not end in T, then, since the geodesic $xz$ varies continuously with z, for some z in the interior of e, the geodesic $xz$ ends in e. Denote by $e_1, e_2$ the two subedges into which such z divides e. Suppose that $xz$ ends in $e_1$ . Then the entire $e_1$ must lie in $xz$ . Moreover, appending $xz$ by $e_2$ , we also obtain a geodesic (Theorem 2.2). Since y lies in $e=e_1\cdot e_2$ , this shows that $xy$ ends in e, which is a contradiction.

Proof. Consider the geodesic triangle $xyz$ . By our assumptions, its Alexandrov angle at z is $>\frac {\pi }{2}$ , and so, in particular, $x\neq y$ , and its Alexandrov angle at y is $<\frac {\pi }{2}$ . Since $zy$ ends in a triangle $T'$ perpendicularly to $e'$ , we have that $xy$ ends in $T'$ , as desired.

Proof of Proposition 8.2

Let $\gamma =\gamma _1\cdot \gamma _2\cdots \gamma _{2k-1}\cdot \gamma _{2k}$ as in Definition 8.1. For $i=1,\ldots , k,$ denote $\gamma _{2i}=y_iz_i$ and denote by $T_i$ the triangle in which $\gamma _{2i-1}$ ends. Let x be the starting point of $\gamma _{1}$ . We prove by induction on $i=1,\ldots , k,$ that x and $z_i$ are distinct and that the geodesic $xz_i$ ends in $T_i$ . The proposition follows from this induction hypothesis applied with $i=k$ .

For $i=1$ , the induction hypothesis follows from Lemma 8.3. Suppose now that we have established it for some $i=m<k$ . Then, by Lemma 8.4, the geodesic $xy_{m+1}$ is nontrivial and ends in $T_{m+1}$ . Thus, by Lemma 8.3, the induction hypothesis holds for $i=m+1$ .

Proof of Proposition 3.2

By Proposition 5.3, we can assume that none of the vertex links of X are unfoldable. Thus, by Proposition 7.4 and Lemma 4.1(i), we can assume that X is piecewise Euclidean and extrational. Let $Z\subset X$ be a thick G-c.s. Note that each patch of X either has no triangle in Z or is contained in Z, in which case, we call it a Z-patch.

Since X is rational, and Z is a G-c.s., there is $q\in \mathbb {N}$ , such that for each Z-patch P and each vertex $v\in \partial P$ , the length of $\mathrm {lk}_v^{\overline P}$ is a multiplicity of $\frac {\pi }{q}$ . For each Z-patch P, we define the homomorphism $\psi '=\psi '(P)\colon H_1(\overline P,\partial P)\to \mathbb {R}/\frac {\pi }{q}\mathbb {Z}$ in the same way as $\psi $ , but replacing $\pi \mathbb {Q}$ by $\frac {\pi }{q}\mathbb {Z}$ . We have $\psi =\psi '$ mod $\pi \mathbb {Q}$ . Since $\psi $ is trivial, the image of $\psi '$ is contained in $\pi \mathbb {Q}/\frac {\pi }{q}\mathbb {Z}$ . Since there are finitely many G-orbits of Z-patches, and since each $H_1(\overline P,\partial P)$ is finitely generated as a $\mathrm {Stab}(P)$ -module, there is $q'\in \mathbb {N}$ , such that the image of each $\psi '$ is contained in $\frac {\pi }{q'}\mathbb {Z}/\frac {\pi }{q}\mathbb {Z}$ . Consequently, for any Z-patch P, any geodesic $xy$ in $\overline P$ disjoint from $\partial P$ , except at its endpoints, that is at angle $\in \frac {\pi }{q'}\mathbb {Z}$ from $\partial P$ at x, is also at angle $\in \frac {\pi }{q'}\mathbb {Z}$ from $\partial P$ at y. Without loss of generality, assume that $q'$ is even.

We need the following variant of the Liouville measure $\mu $ from [Reference Ballmann and BrinBB95, Section 3]. Let S be the set of all the directions $\xi $ at an angle $\theta (\xi )\in \frac {\pi }{q'}\mathbb {Z}\cap (-\frac {\pi }{2},\frac {\pi }{2})$ from a direction normal to E in the links $\mathrm {lk}_x^Z$ for all the points $x\in Z$ that lie in the interior of an edge e of E. The Liouville measure $d\mu (\xi )$ on S is given as $\cos \theta (\xi )dx$ , where $dx$ is the volume element on e. Let $V\subset S$ be the full measure subset of S of directions $\xi $ , such that each geodesic ray $\gamma $ in Z with starting direction $\xi $ is disjoint from $Z^0$ . Let $F\colon V\to \mathcal P(V)$ be the map defined by $\eta \in F(\xi )$ for $\eta \in \mathrm {lk}_x^Z$ if there exists a geodesic $yz$ in Z with starting direction $\xi $ , intersecting E only in y and x and with $\eta $ being the direction at x of $xz$ . Since G acts on Z properly and cocompactly, we have that each $F(\xi )$ is finite. We can thus define a Markov chain with states V and transition probabilities $\frac {1}{|F(\xi )|}$ from $\xi $ to each $\eta \in F(\xi )$ . By (the calculation in) [Reference Ballmann and BrinBB95, Proposition 3.3], the measure $\mu $ is stationary for this Markov chain. Thus, the space $V^{\mathbb {Z}}$ can be equipped with Markov measure $\mu ^*$ invariant under the shift (see, e.g. [Reference WaltersWal82, Example (8), page 21]). Since Z is a G-c.s., the quotient $V^{\mathbb {Z}}/G$ by the diagonal action of G is of finite measure. Note that the shift map descends to $V^{\mathbb {Z}}/G$ and is still measure preserving.

Let e be an edge of Z lying in three distinct triangles $T_a,T_b,T_c$ of Z. Let $V^{ab}\subset V^{\mathbb {Z}}$ be the set of $(\xi _i)_i$ , such that

• • we have $\xi _1\in \mathrm {lk}_x^{T_a}$ for $x\in e$ , and $\xi _1$ is at angle $\frac {\pi }{2}$ from e and

• • the geodesic $yx$ from the definition of $\xi _1\in F(\xi _0)$ ends in $T_b$ .

Note that $V^{ab}$ has positive Markov measure. Thus, by the Poincaré recurrence (see, e.g. [Reference WaltersWal82, Theorem 1.4]), there is $(\xi _i)_i\in V^{ab}$ and $j>0$ with $(\xi _{i-j})_i\in GV^{ab}$ . Consequently, there is a geodesic $\gamma ^{ab}$ in $Z\setminus Z^0$ starting perpendicularly to e in $T_a$ and ending perpendicularly to a translate $fe$ in $fT_b$ , for some $f\in G$ .

Denote by $a, fb$ the endpoints of $\gamma ^{ab}$ . Let $I^{ab}$ be the domain of the isometric embedding $\gamma ^{ab}\colon I^{ab}\to Z\setminus Z^0$ . Analogously, there is a geodesic $\gamma ^{ca}\colon I^{ca}\to Z\setminus Z^0$ starting perpendicularly to e in $T_c$ and ending perpendicularly to a translate $ge$ in $gT_a$ , with endpoints $c, a'$ , for some $g\in G$ . Finally, there is a geodesic $\gamma ^{bc}\colon I^{bc}\to Z\setminus Z^0$ starting perpendicularly to $ge$ in $T_b$ and ending perpendicularly to a translate $f'ge$ in $f'gT_c$ , with endpoints $b', f'c'$ , for some $f'\in G$ . Let $\gamma \colon I\to e$ be the shortest geodesic in e containing all $a,b,c$ in its image (possibly I is a single point), and let $\gamma '\colon I'\to ge$ be the shortest geodesic in $ge$ containing all $a',b',c'$ in its image.

Let $\Gamma $ be the metric graph obtained in the following way. We start from the disjoint union of the five intervals $I^{ab},I^{ca},I^{bc},I,I'$ , and we identify (see Figure 6):

• • points of I and $I^{ab}$ mapping to a under $\gamma $ and $\gamma ^{ab}$ ,

• • points of I and $I^{ca}$ mapping to c under $\gamma $ and $\gamma ^{ca}$ ,

• • points of $I'$ and $I^{ca}$ mapping to $a'$ under $\gamma '$ and $\gamma ^{ca}$ and

• • points of $I'$ and $I^{bc}$ mapping to $b'$ under $\gamma '$ and $\gamma ^{bc}$ .

Note that $\Gamma $ admits the map $\varphi \colon \Gamma \to Z$ that is the quotient of $\gamma ^{ab}\sqcup \gamma ^{ca}\sqcup \gamma ^{bc}\sqcup \gamma \sqcup \gamma '$ . Let $s,t,s',t'$ , be the points in $I,I^{ab},I',I^{bc}$ mapping under $\varphi $ to $b, fb, c',f'c'$ , respectively.

Figure 6

The graph $\Gamma $ .

Let $F_2$ be the free group on two generators $h,h'$ , and let $\widehat \Gamma $ be the quotient of the graph $F_2\times \Gamma $ (which is the disjoint union of $F_2$ copies of $\Gamma $ ) by the relations $w\times t\sim wh\times s, w\times t'\sim wh'\times s'$ , for all $w\in F_2$ . Note that $\widehat \Gamma $ is a tree with a free action of $F_2$ . Let $\varphi _*\colon F_2 \to G$ be the homomorphism mapping $h,h'$ to $f,f'$ , respectively. Then $\varphi $ extends to a $\varphi _*$ -equivariant map $\widehat \varphi \colon \widehat \Gamma \to Z$ mapping each $w\times r\in F_2\times \Gamma $ to $\varphi _*(w)\varphi (r)\in Z$ .

Let w be a nontrivial element of $F_2$ , and let $\mathbb {R}_w$ be the axis for w in $\widehat \Gamma $ . Pick $p\in \mathbb {R}_w$ an endpoint of a translate of one of $I^{ab},I^{ca},I^{bc}$ contained in $\mathbb {R}_w$ . Let $I_w\subset \mathbb {R}_w$ be the interval between p and $wp$ , and let $\gamma _w\colon I_w\to Z$ be the restriction of $\widehat \varphi $ to $I_w$ . Since $T_a,T_b,T_c,$ were distinct, we have that $\gamma _w$ is a sheared geodesic. By Proposition 8.2, we have $\widehat \varphi (p)\neq \widehat \varphi (wp)=\varphi _*(w)\widehat \varphi (p)$ , and, consequently, $\varphi _*(w)$ is nontrivial. Thus, $\varphi _*$ is injective, and so G contains a nonabelian free group.