Henrique Souza
Universidad Autónoma de Madrid
Advisor: Andrei Jaikin-Zapirain
Over a field or division ring
\mathrm{rk}(A) = \dim(\mathrm{im}(A))
Definition: inner rank in a ring R
\mathrm{irk}_R(A) is the smallest k such that A_{m\times n} = B_{m \times k} \cdot C_{k \times n} over R.
Depends on the ring
Over \mathbb{Q}[x^2,x^3] the matrix \begin{pmatrix} x^2 & x^3 \\ x^3 & x^4\end{pmatrix} = \begin{pmatrix}x^2 \\ x^3\end{pmatrix}\begin{pmatrix}1 & x\end{pmatrix} has inner rank 2
Maximal rank doesn’t imply invertible
Over \mathbb{Q}[x,y,z]: \begin{pmatrix}0 & z & -y\\ -z & 0 & x\\ y & -x & 0\end{pmatrix}\begin{pmatrix} x \\ y \\ z\end{pmatrix} = 0
Definition: Sylvester rank functions
Is a function \mathrm{rk}\colon \mathrm{Mat}(R) \to \mathbb{R}_{\geq 0} that satisfies the expected formal properties: \mathrm{rk}(0) = 0, \mathrm{rk}(1) = 1, \mathrm{rk}(AB) \leq \min\{\mathrm{rk}(A),\mathrm{rk}(B)\}, \mathrm{rk}\begin{pmatrix}A & 0 \\ 0 & B\end{pmatrix} = \mathrm{rk}(A) + \mathrm{rk}(B) and \mathrm{rk}\begin{pmatrix} A & C\\ 0 & B\end{pmatrix} \geq \mathrm{rk}(A) + \mathrm{rk}(B).
Examples
Definition: Sylvester domains
Is a ring R such that \operatorname{irk}_R satisfies the Sylvester rank function axioms.
Theorem. (P. Malcolmson, 1980)
Any integer valued rank function \mathrm{rk} \in \mathbb{P}(R) is the pullback of the usual rank over a unique ring epimorphism R \to \mathcal{D} onto a division ring \mathcal{D}.
Definition
A profinite ring is a compact Hausdorff ring. Equivalently, it is an inverse limit of finite discrete rings.
Theorem. (Jaikin, 2020)
Let R be a projective-free ring and \varphi\colon R \to \mathcal{D} be a ring homomorphism to a division ring. Assume that:
Then, R is a Sylvester domain and \varphi^\# \mathrm{rk}_{\mathcal{D}} = \mathrm{irk}_R.
Theorem. (Jaikin-S., 2025)
If G is a torsion-free finitely generated pro-p group, then \mathbb{F}_p[\![G]\!] is a Sylvester domain if and only if \mathbb{F}_p[\![U]\!] is a Sylvester domain for some/any open subgroup U of G.
Fact.
G is f.g. free-by-\mathbb{Z}_p iff G admits a flag presentation: G = \langle x_1,\ldots,x_n,t \mid \mathrm{ad}(t)^{k_i}(x_i) = w_i, i=1,\ldots,n\rangle for some k_i \geq 1 and w_i \in F^p[F,F] where F = \langle x_1,\ldots,x_n\rangle.
Theorem. (Jaikin-S., 2025)
If G is a torsion-free finitely generated virtually free-by-\mathbb{Z}_p pro-p group, then \mathbb{F}_p[\![G]\!] is a Sylvester domain.
Proposition.
Assume G has a mild flag presentation. Under the radical filtration, \mathrm{Gr}(\mathbb{F}_p[\![G]\!]) = \mathbb{F}_p\langle y_1,\ldots,y_n,\ldots\rangle [\overline{t-1};\operatorname{id},\overline{\delta}] where \mathbb{F}_p\langle y_1,\ldots,y_n,\ldots\rangle is the free associative algebra.
Theorem. (Jaikin-S., 2025)
If G = \varprojlim G_i is a torsion-free finitely generated virtually free-by-\mathbb{Z}_p pro-p group, then \mathrm{irk}_{\mathbb{F}_p[\![G]\!]} = \lim \mathrm{rk}_{G_i} = \mathrm{rk}_G\,.
Theorem. (Jaikin-S.,2025)
Let \varphi\colon \mathbb{F}_p[\![G]\!] \to \mathcal{D} be a continuous homomorphism to a division ring \mathcal{D} endowed with a discrete valuation. Then, \varphi^\#\mathrm{rk}_{\mathcal{D}} \leq \mathrm{rk}_G as rank functions on \mathbb{P}(\mathbb{F}_p[\![G]\!]).
Theorem. (Jaikin-S., 2025)
Let R be a profinite Sylvester domain with associated epic universal division R-ring \mathcal{D}. Then, \mathcal{D} admits a Hausdorff ring topology such that the embedding R \to \mathcal{D} is continuous.
Cohomology growth
Let G be a complex semisimple Lie group, \Gamma \leq G be a finitely generated subgroup and V_\lambda be the irreducible rational representation of G with highest weight \lambda. How does the cohomology groups \mathrm{H}^i(\Gamma,V_\lambda) grow as \lambda \to \infty?
Fact.
Let G be a semisimple complex Lie group. The irreducible rational representations of G are classified by their highest weight \lambda \in \mathfrak{h}^*, where \mathfrak{h} is a Cartan subalgebra of \operatorname{Lie}(G).
Proposition.
For \Gamma of type FP_\infty(\mathbb{C}) there are fixed matrices A and B over \mathbb{C} and an integer constant C such that for any complex representation \varphi\colon \Gamma \to \mathsf{GL}(V), \dim_{\mathbb{C}} \mathrm{H}^i(\Gamma, V) = \dim_{\mathbb{C}}(V)(C - \mathrm{rk}_{\varphi}(A) - \mathrm{rk}_{\varphi}(B))\,.
Theorem. (Guerrero-S., submitted)
Let G be a quotient of \prod_{i=1}^k \mathsf{SL}_2(\mathbb{C}) by a central subgroup and \Gamma \leq G be a finitely generated torsion-free subgroup. Then, for the rational representations V_\lambda, \mathrm{rk}_{\varphi_\lambda} \to \mathrm{rk}_\Gamma \text{ as }\min_i \lambda_i \to \infty\,.
Corollary.
Let G be a quotient of \prod_{i=1}^k \mathsf{SL}_2(\mathbb{C}) by a central subgroup and \Gamma \leq G be a torsion-free subgroup of type FP_\infty(\mathbb{C}). Then: \frac{\dim_{\mathbb{C}} \mathrm{H}^i(\Gamma,V_\lambda)}{\dim_{\mathbb{C}} V_\lambda} \to \beta_i^{(2)}(\Gamma) as \min_j \lambda_j \to \infty.
Theorem. (Cassels, 1976)
If \Gamma is a f.g. subgroup of G, there are infinitely many primes p such that \Gamma embeds in G(\mathbb{Z}_p) = \prod_{i=1}^k \mathsf{SL}_2(\mathbb{Z}_p)\,.
Facts.
For odd primes p, the Iwasawa algebra \mathbb{Z}_p[\![U_1]\!] is a Noetherian domain. The pullback of the Ore rank function on \mathbb{Z}_p[\![U_1]\!] to S[\Gamma \cap U_1] coincides with the von Neumann rank.
Conjecture. (Guerrero-S., submitted)
Let U be a compact uniform split semisimple and simply-connected p-adic Lie group, V_\lambda be an irreducible p-adic Lie representation of U and \mathbb{Z}_p[\![U]\!] be its Iwasawa algebra. For any non-zero x \in \mathbb{Z}_p[\![U]\!] we have \frac{\dim_{\mathbb{Q}_p} xV_\lambda}{ \dim_{\mathbb{Q}_p} V_\lambda} = 1 as \min_i \lambda_i \to \infty.
Theorem. (Ardakov-Wadsley, 2014)
For almost all primes p, the composition \mathbb{Z}_p[\![U]\!] \to \widehat{U(\mathfrak{g})} \to \widehat{U(\mathfrak{g})}/I_\lambda is injective.