2 Expander representations and uniform expansion

For all general notions about quivers and their representations, we refer to [Reference Schiffler16]. Let Q be a finite quiver, given by a finite set $Q_0$ of vertices and finitely many arrows written $\alpha :i\rightarrow j$ . We assume Q to be acyclic, that is, Q has no oriented cycles. We consider the real vector space $\mathbb {R}Q_0$ with coordinate basis vectors $\mathbf {i}$ for $i\in Q_0$ , and write vectors $\mathbf {d}\in \mathbb {R}Q_0$ as

$$ \begin{align*}\mathbf{d}=\sum_{i\in Q_0}d_i\mathbf{i}.\end{align*} $$

We define the Euler form of Q as the bilinear form

$$ \begin{align*}\langle\mathbf{d},\mathbf{e}\rangle=\sum_{i\in Q_0}d_ie_i-\sum_{\alpha:i\rightarrow j}d_ie_j.\end{align*} $$

We denote by $(\_,\_)$ its symmetrization, and by $\{\_,\_\}$ its antisymmetrization. With respect to the coordinate basis, $(\_,\_)$ is represented by a matrix C, which is a symmetric generalized Cartan matrix.

We fix an algebraically closed base field F and consider finite-dimensional F-representations V of Q. Thus, V is given by finite-dimensional F-vector spaces $V_i$ for $i\in Q_0$ and F-linear maps $V_\alpha :V_i\rightarrow V_j$ for all arrows $\alpha :i\rightarrow j$ . Morphisms between representations V and W are tuples $(\varphi _i:V_i\rightarrow W_i)_{i\in Q_0}$ of linear maps intertwining the structure maps, that is,

$$ \begin{align*}W_\alpha\circ \varphi_i=\varphi_j\circ V_\alpha\end{align*} $$

for all arrows $\alpha :i\rightarrow j$ in Q. Defining composition of morphisms componentwise results in an abelian F-linear category $\textrm {rep}_FQ$ of finite-dimensional F-representations of Q, which is equivalent to the category of finite-dimensional left modules over the path algebra $FQ$ of Q. In particular, we can apply all standard concepts of the theory of modules over rings to quiver representations. The dimension vector of a representation V is the vector

$$ \begin{align*}\mathbf{dim}(V)=\sum_{i\in Q_0}(\dim_FV_i)\mathbf{i}\in\mathbb{R}_{\geq 0}Q_0.\end{align*} $$

We next introduce stability and refer to [Reference Hille and de la Peña5, Reference King8, Reference Rudakov15] for generalities. Let $\Theta ,\kappa \in (\mathbb {R}Q_0)^*$ be two linear functionals, such that $\kappa $ is positive in the sense that

$$ \begin{align*}\kappa(\mathbb{R}_{\geq 0}Q_0\setminus\{0\})\subset\mathbb{R}_{>0}\end{align*} $$

(equivalently, $\kappa (\mathbf {i})>0$ for all $i\in Q_0$ ). We define the associated slope function

$$ \begin{align*}\mu(\mathbf{d})=\Theta(\mathbf{d})/\kappa(\mathbf{d})\end{align*} $$

for $\mathbf {d}\in \mathbb {R}_{\geq 0}Q_0\setminus \{0\}$ . We extend the functional $\Theta $ to representations by

$$ \begin{align*}\Theta(V)=\Theta(\mathbf{dim}(V)),\end{align*} $$

and similarly for $\kappa $ and for the slope function $\mu $ .

We call a nonzero representation V stable with respect to the slope function $\mu $ , or shortly $\mu $ -stable, if $\mu (U)<\mu (V)$ for all nonzero proper subrepresentations $U\subset V$ . Note that the class of stable representations does not change when $\Theta ,\kappa $ are replaced by positive scalar multiples, or when $\Theta $ is shifted by a multiple of $\kappa $ . Stable representations V are Schurian, that is, they have trivial endomorphism ring $\textrm {End}_Q(V)\simeq F$ , by a Schur’s-lemma-type argument.

From now on, we will fix a slope function $\mu $ . We can now proceed to our main new concept.

The precise relation to stability is easily derived:

Proof. Assume that the second condition is fulfilled, and let $U\subset V$ be a nonzero proper subrepresentation. Then $\kappa (U)=\delta \cdot \kappa (V)$ for some $\delta \in ]0,1[$ ; thus

$$ \begin{align*}\mu(U)\leq\mu(V)-\epsilon(\delta)<\mu(V),\end{align*} $$

thus V is $\mu $ -stable. Conversely, assume that V is $\mu $ -stable, and fix $\delta \in ]0,1[$ . If V is simple, there is nothing to prove. Otherwise, there are only finitely many dimension vectors $\mathbf {e}$ of subrepresentations of V such that $\kappa (\mathbf { e})=\delta \cdot \kappa (V)$ ; thus

$$ \begin{align*}\epsilon(\delta)=\min\{\mu(V)-\mu(U)\, |\, 0\not=U\subset V,\, \kappa(U)\leq\delta\cdot\kappa(V)\}\end{align*} $$

exists and is positive by $\mu $ -stability of V. Thus, the second condition is fulfilled, finishing the proof.

More important than individual expander representations are families of expanders for which the bound $\epsilon =\epsilon (\delta )$ can be chosen uniformly for the whole family:

Our main result is:

This will be proved in Section 7.

3 Numerical criterion

We first recall methods and results from [Reference Schofield17].

For two dimension vectors $\mathbf {e}\leq \mathbf {d}$ of representations of Q, we write $\mathbf {e}\hookrightarrow \mathbf {d}$ if every representation of dimension vector $\mathbf {d}$ admits a subrepresentation of dimension vector $\mathbf {e}$ . This condition can be characterized numerically, although recursively [Reference Crawley-Boevey2, Reference Schofield17]:

For a given dimension vector $\mathbf {d}$ and fixed vector spaces $V_i$ of dimension $d_i$ for $i\in Q_0$ , respectively, the representations V of Q on the spaces $V_i$ are naturally parametrized by the affine space

$$ \begin{align*}R_{\mathbf{d}}(Q)=\bigoplus_{\alpha:i\rightarrow j}\textrm{Hom}_F(V_i,V_j),\end{align*} $$

which we consider with its Zariski topology. For another dimension vector $\mathbf {e}\leq \mathbf {d}$ , the subset $S_{\mathbf {e}}\subset R_{\mathbf {d}}(Q)$ of representations admitting a subrepresentation of dimension vector $\mathbf {e}$ is known to be Zariski-closed. By definition, we have $\mathbf {e}\hookrightarrow \mathbf {d}$ if and only if $S_{\mathbf {e}}=R_{\mathbf {d}}(Q)$ .

The following is the key to existence proofs for (uniform) expanders. The proof is nonconstructive and requires F to be algebraically closed, but it effectively reduces existence to a purely numerical/root-theoretic problem.

Proof. Let V be a $(\delta ,\epsilon )$ -expander of dimension vector $\mathbf {d}$ , and let $0\not =\mathbf {e}\hookrightarrow \mathbf {d}$ be a dimension vector such that $\kappa (\mathbf {e})\leq \delta \cdot \kappa (\mathbf {d})$ . Then there exists a subrepresentation $U\subset V$ of dimension vector $\mathbf {e}$ , for which $\kappa (U)\leq \delta \cdot \kappa (V)$ , and thus

$$ \begin{align*}\mu(\mathbf{e})=\mu(U)\leq\mu(V)-\epsilon=\mu(\mathbf{d})-\epsilon.\end{align*} $$

Conversely, assume that the numerical condition holds. Consider the (finite) set D of all dimension vectors $0\not =\mathbf {e}\leq \mathbf {d}$ such that $\kappa (\mathbf {e})\leq \delta \cdot \kappa (\mathbf {d})$ , but $\mu (\mathbf {e})>\mu (\mathbf { d})-\epsilon $ . For every $\mathbf {e}$ in D, the condition $\mathbf {e}\hookrightarrow \mathbf {d}$ does not hold by assumption; thus $S_{\mathbf {e}}\subset R_{\mathbf {d}}(Q)$ is a proper Zariski-closed subset. Thus, the finite union

$$ \begin{align*}\bigcup_{\mathbf{e}\in D}S_{\mathbf{e}}\subset R_{\mathbf{d}}(Q)\end{align*} $$

is still a proper closed subset, and we can choose a representation V in its complement. If $U\subset V$ is a subrepresentation such that $\kappa (U)\leq \delta \cdot \kappa (V)$ , and $\mathbf {e}=\mathbf {dim}(U)$ , then $\mathbf {e}$ does not belong to D by the choice of V. Thus

$$ \begin{align*}\mu(U)=\mu(\mathbf{e})\leq\mu(V)-\epsilon,\end{align*} $$

proving that V is a $(\delta ,\epsilon )$ -expander.

This lemma motivates the following notation:

Definition 3.3. Given a dimension vector $\mathbf {d}$ and $\delta \in ]0,1[$ , define

$$ \begin{align*}\epsilon_{\mathbf{d}}^{\mathrm{opt}}(\delta)=\min\{\mu(\mathbf{d})-\mu(\mathbf{e})\, |\, 0\not=\mathbf{e}\hookrightarrow\mathbf{d},\, \kappa(\mathbf{e})\leq\delta\cdot\kappa(\mathbf{d})\}\end{align*} $$

and

$$ \begin{align*}\epsilon_{\mathbf{d}}^{\mathrm{eff}}(\delta)=\min\{\mu(\mathbf{d})-\mu(\mathbf{e})\, |\, 0\not=\mathbf{e}\leq\mathbf{d},\, \langle\mathbf{e},\mathbf{d}-\mathbf{e}\rangle\geq 0,\, \kappa(\mathbf{e})\leq\delta\cdot\kappa(\mathbf{d})\}.\end{align*} $$

Thus, by the previous lemma, $\epsilon _{\mathbf {d}}^{\mathrm {opt}}(\delta )$ is the largest $\epsilon $ for which a $(\delta ,\epsilon )$ -expander of dimension vector $\mathbf {d}$ exists. It is difficult to determine, though, due to the recursive nature of the criterion for deciding $\mathbf {e}\hookrightarrow \mathbf {d}$ (see, however, Section 4 for the case of generalized Kronecker quivers). In contrast, since $\mathbf {e}\hookrightarrow \mathbf {d}$ implies $\langle \mathbf {e},\mathbf {d}-\mathbf {e}\rangle \geq 0$ , we have

$$ \begin{align*}\epsilon_{\mathbf{d}}^{\mathrm{eff}}(\delta)\leq\epsilon^{\mathrm{opt}}_{\mathbf{d}}(\delta),\end{align*} $$

and this value is sometimes easier to determine. This motivates our strategy for proving existence of uniform expanders:

4 The case of generalized Kronecker quivers

Before treating the general case, we use results of [Reference Reineke12] to obtain rather complete results on uniform expansion for generalized Kronecker quivers. So assume $m\geq 3$ , and let Q be the quiver with $Q_0=\{1,2\}$ and m arrows from $1$ to $2$ . Let $\mathbf {d}=d_1\mathbf {1}+d_2\mathbf {2}$ be a dimension vector such that $(\mathbf {d},\mathbf { d})=d_1^2-md_1d_2+d_2^2<0$ .

Proof. By [Reference Reineke12, Proposition 3.4], we have $\mathbf {e}\hookrightarrow \mathbf {d}$ if and only if $\langle \mathbf {e},\mathbf {d}-\mathbf {e}\rangle \geq 0$ if and only if $e_2\geq c_{\mathbf {d}}(e_1)$ , where the function $c_{\mathbf {d}}(x)$ is given by

$$ \begin{align*}c_{\mathbf{d}}(x)=\frac{1}{2}(mx+d_2-\sqrt{(mx-d_2)^2+4x(d_1-x)}).\end{align*} $$

Moreover, the proof of [Reference Reineke12, Theorem 4.3] shows that, for any $0<x<d_1$ , we can write

$$ \begin{align*}c_{\mathbf{d}}(x)=(1+\zeta_{\alpha}(x/d_1))\cdot\frac{d_2}{d_1}\cdot x\end{align*} $$

for a positive $\zeta _{\alpha }(x/d_1)$ , where $\zeta _{\alpha }(x/d_1)$ , as a function of x, is strictly monotonically decreasing and only depends on $\alpha =d_2/d_1$ , and not on $\mathbf {d}$ .

Let $\mu =\Theta /\kappa $ be a stability for Q. We can assume that $\mu (\mathbf {d})=0$ , thus, without loss of generality, $\Theta (\mathbf {1})=d_2$ , $\Theta (\mathbf {2})=-d_1$ . Using nonnegativity of $\mathbf {d}$ , $\mathbf {e}$ and $\kappa $ , for any $0\not =\mathbf {e}\hookrightarrow \mathbf {d}$ we can then estimate

$$ \begin{align*}-\mu(\mathbf{e})=\frac{-d_2e_1+d_1e_2}{\kappa_1e_1+\kappa_2e_2}\geq\frac{-d_2e_1+d_1c_{\mathbf{d}}(e_1)}{\kappa_1e_1+\kappa_2c_{\mathbf{d}}(e_1)}=\frac{-d_2e_1+(1+\zeta_\alpha(e_1/d_1))d_2e_1}{\kappa_1e_1+\kappa_2(1+\zeta_\alpha(e_1/d_1))d_2e_1/d_1}\end{align*} $$

$$ \begin{align*}=\frac{d_1d_2\cdot\zeta_\alpha(e_1/d_1)}{(\kappa_1d_1+\kappa_2d_2)+\kappa_2d_2\cdot\zeta_{\alpha}(e_1/d_1)}.\end{align*} $$

If $\kappa (\mathbf {e})=\delta \cdot \kappa (\mathbf {d})$ , we can estimate

$$ \begin{align*}\delta\cdot(\kappa_1d_1+\kappa_2d_2)=\kappa_1e_1+\kappa_2e_2>\kappa_1e_1+\kappa_2d_2e_1/d_1=(\kappa_1d_1+\kappa_2d_2)e_1/d_1,\end{align*} $$

and thus $e_1/d_1<\delta $ . This allows us to finally bound $-\mu (\mathbf {e})$ from below by a term depending only on $\delta $ (and $\alpha $ ), namely

$$ \begin{align*}-\mu(\mathbf{e})\geq \frac{d_1d_2\cdot\zeta_{\alpha}(\delta)}{\kappa(\mathbf{d})+\kappa_2d_2\cdot\zeta_{\alpha}(\delta)}=:\epsilon_{\mathbf{d}}(\delta).\end{align*} $$

Application of Lemma 3.4 finishes the proof.

Remark. Essentially the same calculation shows that a $(\delta ,\epsilon )$ -expander in the sense of [Reference Reineke12, Definition 4.1] is a $(\delta ',\epsilon ')$ -expander representation with respect to the above slope function, for

$$ \begin{align*}\delta'=\frac{\delta\cdot\kappa_1d_1+\kappa_2d_2}{\kappa_1d_1+\kappa_2d_2},\; \epsilon'=\frac{d_1d_2\cdot\epsilon}{\kappa(\mathbf{d})+\kappa_2d_2\cdot\epsilon}.\end{align*} $$

5 Construction of uniform expanders for wild quivers

Before applying Lemma 3.4 in the generality of arbitrary wild quivers, we recall properties of the Cartan matrix C and their relation to representations of Q, mainly following [Reference Kac6, Reference Kac7].

We can assume the quiver Q to be connected. The Cartan matrix C of Q is then irreducible (that is, it cannot be transformed to proper block-diagonal form by a reindexing of $Q_0$ ). It is positive definite if and only if Q is of Dynkin type, positive semi-definite if and only if Q is of Dynkin or extended Dynkin type, and indefinite in all other cases, in which Q is called wild.

Denote by n the cardinality of $Q_0$ . By definition, the matrix $2\cdot E_n-C$ has nonnegative entries and is still irreducible; thus the Perron-Frobenius theorem applies to it. Together with the symmetry of C, this immediately yields:

Here we used the standard Euclidean scalar product in $\mathbb {R}Q_0$ , for which the coordinate elements $\mathbf {i}$ form an orthonormal basis, $\mathbf {i}\cdot \mathbf {j}=\delta _{ij}$ .

The so-called fundamental domain of $\mathbb {R}_{>0}Q_0$ is the cone of all $\mathbf {d}\in \mathbb {R}_{>0}Q_0$ for which $(\mathbf { d},\mathbf {i})\leq 0$ for all $i\in Q_0$ (it is a fundamental domain for the Weyl group action on imaginary roots for the Cartan matrix C). By [Reference Kac7, Lemma 1] and [Reference Schofield17, Theorem 6.1], we have:

We now start with the main construction of a uniform expander. Let Q be wild and connected, and let $\mathbf {d}\in \mathbb {R}_{>0}Q_0$ be a dimension vector in the interior of the fundamental domain, that is, $(\mathbf {d},\mathbf {i})<0$ for all $i\in Q_0$ .

Then $\kappa =-(\mathbf {d},\_)$ is a positive functional; we define $\Theta =\{\mathbf {d},\_\}$ , and we denote by $\mu $ the associated slope function. By Theorem 5.2, we then know that $\mathbf {d}$ admits a Schurian representation, thus a $\mu $ -stable representation. Note that any positive multiple of $\mathbf {d}$ gives the same slope function $\mu $ , and also admits a $\mu $ -stable representation. The family $(k\mathbf {d})_k$ is our candidate for a uniform expander, and we would like to apply Lemma 3.4. The following lemma provides the key technical estimate.

Lemma 5.3. There exists an open neighborhood $\mathcal {U}\subset \mathbb {R}_{>0}Q_0$ of the ray $\mathbb {R}_{>0}\mathbf {v}$ such that, for all $\mathbf {d}\in \mathcal {U}$ ,

there exists a positive constant C with the following property: if $\mathbf {e}\leq \mathbf {d}$ satisfies $\langle \mathbf {e},\mathbf { d}-\mathbf {e}\rangle \geq 0$ , then

$$ \begin{align*}\{\mathbf{d},\mathbf{e}\}\leq C\cdot\eta\cdot(1-\eta)\cdot(\mathbf{d},\mathbf{d}),\end{align*} $$

where $\kappa (\mathbf {e})=\eta \cdot \kappa (\mathbf {d})$ .

Proof. We have

$$ \begin{align*}\{\mathbf{d},\mathbf{e}\}=\langle\mathbf{d},\mathbf{e}\rangle-\langle\mathbf{e},\mathbf{d}\rangle,\; (\mathbf{d},\mathbf{e})= \langle\mathbf{d},\mathbf{e}\rangle+\langle\mathbf{e},\mathbf{d}\rangle,\end{align*} $$

thus the assumption on $\mathbf {e}$ can be reformulated as

$$ \begin{align*}0\leq\langle\mathbf{e},\mathbf{d}-\mathbf{e}\rangle=\langle\mathbf{e},\mathbf{d}\rangle-\langle\mathbf{e},\mathbf{e}\rangle=\frac{1}{2}(\mathbf{d},\mathbf{e})-\frac{1}{2}\{\mathbf{d},\mathbf{e}\}-\frac{1}{2}(\mathbf{e},\mathbf{e}),\end{align*} $$

which implies

$$ \begin{align*}\{\mathbf{d},\mathbf{e}\}\leq(\mathbf{d},\mathbf{e})-(\mathbf{e},\mathbf{e}).\end{align*} $$

By the definition of $\kappa $ and of $\eta $ , we can write uniquely

$$ \begin{align*}\mathbf{e}=\eta\cdot\mathbf{d}+\mathbf{x}\end{align*} $$

for an element $\mathbf {x}\in \mathbb {R}Q_0$ such that

$$ \begin{align*}(\mathbf{d},\mathbf{x})=0,\; -\eta\cdot\mathbf{d}\leq\mathbf{x}\leq(1-\eta)\cdot\mathbf{d}.\end{align*} $$

Then

$$ \begin{align*}(\mathbf{d},\mathbf{e})=\eta\cdot(\mathbf{d},\mathbf{d}),\; (\mathbf{e},\mathbf{e})=\eta^2\cdot(\mathbf{d},\mathbf{d})+(\mathbf{ x},\mathbf{x}),\end{align*} $$

and thus

$$ \begin{align*}\{\mathbf{d},\mathbf{e}\}\leq\eta\cdot(\mathbf{d},\mathbf{d})-\eta^2\cdot(\mathbf{d},\mathbf{d})-(\mathbf{x},\mathbf{x})=\eta\cdot(1-\eta)\cdot(\mathbf{d},\mathbf{d})-(\mathbf{x},\mathbf{x}).\end{align*} $$

If $(\mathbf {x},\mathbf {x})\geq 0$ , this estimate proves the claim for $C=1$ without further restrictions on $\mathbf {d}$ . So assume $(\mathbf {x},\mathbf {x})<0$ , and write $(\mathbf {x},\mathbf {x})=\lambda \cdot \mathbf {x}\cdot \mathbf {x}$ for negative $\lambda $ .

We derive an estimate for $\mathbf {x}\cdot \mathbf {x}$ . From the bounds $-\eta d_i\leq x_i\leq (1-\eta )d_i$ , we first find $(x_i+\eta d_i)\cdot ((1-\eta )d_i-x_i)\geq 0$ , and thus

$$ \begin{align*}x_i^2\leq\eta(1-\eta)d_i^2+(1-2\eta)x_id_i\end{align*} $$

for all $i\in Q_0$ . Defining $w_i=-(\mathbf {d},\mathbf {i})>0$ , we then estimate

$$ \begin{align*}\mathbf{x}\cdot\mathbf{x}&=\sum_{i\in Q_0}\frac{d_i}{w_i}\frac{w_i}{d_i}x_i^2\leq \max\{\frac{d_i}{w_i}|i\in Q_0\}\cdot\sum_{i\in Q_0}\frac{w_i}{d_i}x_i^2\leq\\ &\leq\max\{\frac{d_i}{w_i}|i\in Q_0\}\cdot \sum_{i\in Q_0}\frac{w_i}{d_i}\left(\eta(1-\eta)d_i^2-(1-2\eta)x_id_i\right)\\ &=\max\{\frac{d_i}{w_i}|i\in Q_0\}\cdot(\eta(1-\eta)\underbrace{\sum_{i\in Q_0}w_id_i}_{=-(\mathbf{d},\mathbf{ d})}-(1-2\eta)\underbrace{\sum_{i\in Q_0}w_ix_i}_{=-(\mathbf{d},\mathbf{x})=0})\\ &=-\max\{\frac{d_i}{w_i}|i\in Q_0\}\cdot\eta(1-\eta)\cdot(\mathbf{d},\mathbf{d}). \end{align*} $$

This implies (since $\lambda <0$ by assumption)

$$ \begin{align*}(\mathbf{x},\mathbf{x})=\lambda\cdot\mathbf{x}\cdot\mathbf{x}\geq -\lambda\cdot\max\{\frac{d_i}{w_i}|i\in Q_0\}\cdot\eta(1-\eta)\cdot(\mathbf{d},\mathbf{d}),\end{align*} $$

which yields the estimate

$$ \begin{align*}\{\mathbf{d},\mathbf{e}\}\leq\eta\cdot(1-\eta)\cdot(\mathbf{d},\mathbf{d})-(\mathbf{x},\mathbf{x})\leq\underbrace{(1+\lambda\cdot\max\{\frac{d_i}{w_i}|i\in Q_0\})}_{=:C}\cdot\eta(1-\eta)\cdot(\mathbf{d},\mathbf{d}).\end{align*} $$

Now we use the statements on infima in Lemma 5.1. For $\mathbf {d}$ in a small open neighborhood $\mathcal {U}$ of $\mathbb {R}_{>0}\mathbf {v}$ , the maximum $M:=\max \{\frac {d_i}{w_i}|i\in Q_0\}$ is arbitrarily close to $-\frac {1}{\lambda _1}$ . On the other hand, since $(\mathbf {d},\mathbf {x})=0$ (and $-\eta \cdot \mathbf {d}\leq \mathbf {x}\leq (1-\eta )\cdot \mathbf {d}$ ), the vector $\mathbf { x}$ then belongs to (a bounded region of) an arbitrarily small neighborhood of the hyperplane $\mathbf {v}^\perp $ ; thus $\lambda $ is arbitrarily close to $\lambda _2>\lambda _1$ . Consequently, the constant C is arbitrarily close to $1-\frac {\lambda _2}{\lambda _1}$ , thus in particular positive for small enough $\mathcal {U}$ , proving the claim.

Proof. Indeed, for $0\not =\mathbf {e}\leq \mathbf {d}$ such that $\langle \mathbf {e},\mathbf {d}-\mathbf {e}\rangle \geq 0$ and $\kappa (\mathbf {e})\leq \delta \cdot \kappa (\mathbf {d})$ , we have (note again that $(\mathbf {d},\mathbf {d})<0$ )

$$ \begin{align*}\mu(\mathbf{d})-\mu(\mathbf{e})=\frac{\{\mathbf{d},\mathbf{e}\}}{(\mathbf{d},\mathbf{e})}=\frac{\{\mathbf{d},\mathbf{e}\}}{\eta\cdot(\mathbf{d},\mathbf{d})}\geq\frac{C\cdot\eta\cdot(1-\eta)\cdot(\mathbf{d},\mathbf{d})}{\eta\cdot(\mathbf{d},\mathbf{d})}=C\cdot(1-\eta)\geq C\cdot(1-\delta),\end{align*} $$

proving the first claim. The second claim follows from Lemma 3.4.

6 Nonexistence of preprojective/preinjective uniform expanders

To complement our existence result from the previous section, we prove:

We first recall the definition of preprojective/preinjective representations, in particular limit properties obtained in [Reference Dlab and Ringel4, Reference de la Peña and Takane3, Reference Ringel13, Reference Rudakov15]. We assume that Q is not of Dynkin type and connected. The quiver Q being acyclic, its path algebra $FQ$ is finite-dimensional, and the regular representation $FQ$ decomposes into indecomposable projective ones $P_i$ for $i\in Q_0$ , such that $P_i$ is spanned by all paths starting in i. Similarly, the linear dual $(FQ)^*$ decomposes into injective indecomposable representations $I_i$ for $i\in Q_0$ . Considering $FQ$ and $(FQ)^*$ as bimodules over $FQ$ , we consider the endofunctors

$$ \begin{align*}\tau={\mathrm{Ext}}^1_{FQ}(\_,FQ)^*,\; \tau^{-1}={\mathrm{Ext}}^1_{FQ}((FQ)^*,\_)\end{align*} $$

on left $FQ$ -modules, called (inverse) Auslander-Reiten translation. The $\tau ^{-k}P_i$ for $k\geq 0$ and $i\in Q_0$ , as well as the $\tau ^kI_i$ for $k\geq 0$ and $i\in Q_0$ , form infinite series of Schurian representations, called preprojective (resp. preinjective) indecomposables. One special feature of this family is that every subrepresentation of a preprojective indecomposable $\tau ^{-k}P_i$ is isomorphic to a direct sum of preprojective indecomposables of the form $\tau ^{-l}P_j$ for $l\leq k$ (and dually for factor representations of preinjective ones).

The effect of $\tau ^{(-1)}$ on dimension vectors is described by the Coxeter transformation $\Phi ^{(-1)}\in {\mathrm {GL}}(\mathbb {Z}Q_0)$ . By [Reference Ringel14], the spectral radius $\rho $ of $\Phi $ is at least one (and equal to one if and only if Q is of extended Dynkin type, since Q is assumed not to be of Dynkin type), and it is an eigenvalue of $\Phi $ , as is $\rho ^{-1}$ . For wild quivers Q, by [Reference de la Peña and Takane3], there exist eigenvectors $\mathbf { y}^{\pm }\in \mathbb {R}_{>0}Q_0$ of $\Phi $ for the eigenvalues $\rho ^{\pm 1}$ , respectively, such that

$$ \begin{align*}\lim_{n\rightarrow\infty}\frac{1}{\rho^n}{\mathbf{dim}}(\tau^{-n}P_i)=\lambda^-\cdot\mathbf{y}^-,\;\lim_{n\rightarrow\infty}\frac{1}{\rho^n}{\mathbf{dim}}(\tau^{n}I_i)=\lambda^+\cdot\mathbf{y}^+\end{align*} $$

for positive reals $\lambda ^+,\lambda ^-$ . For Q of extended Dynkin type, by the definition of the defect function in [Reference Dlab and Ringel4, Section 1], we can still choose $\mathbf {y}^+=\mathbf {y}^-\in \mathbb {R}_{>0}Q_0$ to be an eigenvector of $\Phi $ for the eigenvalue $1$ , such that

$$ \begin{align*}\lim_{n\rightarrow\infty}\frac{1}{n}\mathbf{dim}(\tau^{-n}P_i)=\lambda^-\cdot\mathbf{y}^-,\; \lim_{n\rightarrow\infty}\frac{1}{n}\mathbf{dim}(\tau^{n}I_i)=\lambda^+\cdot\mathbf{y}^+\end{align*} $$

for positive reals $\lambda ^+,\lambda ^-$ .

We can now prove Proposition 6.1. Assume that $(V^{(k)})_{k\geq 1}$ is an unbounded family of preprojective or preinjective representations. We can choose an unbounded subfamily consisting entirely of one of the types of representations. We only consider the case of all $V^{(k)}$ being preprojective; the case of preinjective representations can be treated dually. We thus write

$$ \begin{align*}V^{(k)}=\tau^{-n(k)}P_{i(k)},\;\;\; \mathbf{d}^k=\mathbf{dim}(V^{(k)}).\end{align*} $$

Let $\mu =\Theta /\kappa $ be a slope function for which all $V^{(k)}$ are $\mu $ -stable, and fix $\delta \in ]0,1[$ and $\epsilon>0$ . Let i be a sink in Q. Then $P_i$ is a simple representation; thus any nonzero map from $P_i$ to a preprojective representation V is injective. We have

$$ \begin{align*}\mathrm{Hom}(P_i,V)\simeq V_i,\end{align*} $$

thus $P_i$ embeds into V if $\mathbf {dim}(V)_i\not =0$ . Applying $\tau ^{-1}$ , we find that $\tau ^{-n}P_i$ embeds into a preprojective V if $\mathbf {dim}(\tau ^nV)_i\not =0$ . We use the above statements on limits and two properties of the slope function $\mu $ : it is continuous and insensitive to real multiples of a dimension vector. This implies

$$ \begin{align*}\lim_{n\rightarrow\infty}\mu(\tau^{-n}V)=\mu(\mathbf{y}^-)\end{align*} $$

for all preprojective V. We can thus choose n large enough so that

$$ \begin{align*}|\mu(\tau^{-n}P_i)-\mu(\mathbf{y}^-)|<\epsilon/2.\end{align*} $$

With the same argument, and again using the growth behavior of dimension vectors under $\tau ^{-1}$ , we can also find an index k large enough such that

• • $(\Phi ^{n}\mathbf {d}^k)_i\not =0$ , so that $\tau ^{-n}P_i$ embeds into $V^{(k)}$ ,

• • $\kappa (\tau ^{-n}P_i)\leq \delta \cdot \kappa (\mathbf {d}^k)$ ,

• • $|\mu (\mathbf {d}^k)-\mu (\mathbf {y}^-)|<\epsilon /2$ .

We then find (since $V^{(k)}$ is $\mu $ -stable) that

$$ \begin{align*}\mu(V^{(k)})-\mu(\tau^{-n}P_i)=|\mu(V^{(k)})-\mu(\tau^{-n}P_i)|<\epsilon.\end{align*} $$

This contradicts uniform expansion, proving the proposition.

7 Proof of the main result and concluding remarks

We can now prove Theorem 2.4. If Q is wild, we can choose a dimension vector $\mathbf {d}$ in an open neighborhood $\mathcal {U}$ of $\mathbb {R}_{>0}\mathbf {v}$ provided by Lemma 5.3, so that Corollary 5.4 provides a uniform expander supported by the sequence $(k\cdot \mathbf {d})_k$ . If Q is Dynkin, there are only finitely many indecomposable, in fact Schurian, representations up to isomorphism; thus there are no unbounded families of representations at all. If Q is extended Dynkin, by [Reference Dlab and Ringel4, end of Section 3], all Schurian representations are preprojective or preinjective, or have dimension vector at most $\mathbf {y}^-$ . Unbounded families thus have to be preprojective/preinjective, and Proposition 6.1 prevents existence of uniform expanders, finishing the proof of the theorem.