The $n$-$(n+1)$-$(n+2)$ theorem, recently established by Cohen and Shusterman, says that if two groups of type $\mathrm{F}_{n+1}$ map onto a common quotient $Q$ of type $\mathrm{F}_{n+2}$ and one of the two kernels is of type $\mathrm{F}_n$, then the associated fibre product is of type $\mathrm{F}_{n+1}$. We prove a stronger, symmetric variant of this theorem when $Q$ is virtually nilpotent. More precisely, for $i\in\{1,2\}$, let $1\to N_i\to \Gamma_i\overset{\small\pi_i}{\to} Q\to 1$ be short exact sequences of groups and let $P=\{(\gamma_1,\gamma_2)\in\Gamma_1\times\Gamma_2\colon \pi_1(\gamma_1)=\pi_2(\gamma_2)\}$ be their fibre product. Let $k,l,m,n\in\mathbb N_0$, assume that $N_1$ and $N_2$ are of type $\mathrm{FP}_k$ and $\mathrm{FP}_l$, respectively, and that $\Gamma_1$ and $\Gamma_2$ are of type $\mathrm{F}_m$ and $\mathrm{F}_n$, respectively. We prove that $P$ is of type $\mathrm{F}_{\min\{k+l+1,m,n\}}$. The same formula holds with the homological finiteness properties $\mathrm{FP}_r$ in place of $\mathrm{F}_r$. This provides an alternative proof of the homotopical and homological Virtual Surjections Theorem, which gives a finiteness criterion for subgroups of direct products in terms of their embedding in the ambient product.
Let $R$ be a Noetherian standard $\mathbb{N}$-graded ring, and let $I_1,\ldots,I_r$ be homogeneous ideals of $R$. Let $L,M$, and $N$ be finitely generated $\mathbb{Z}$-graded $R$-modules with $N\subseteq M$. For $\underline{n} :=(n_1,\dots,n_r)\in\mathbb N^r$, set ${\bf I}^{\underline{n}}:=I_1^{n_1}\cdots I_r^{n_r}$. W...
Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d$, residue field $k$ and let $I$ be an $\mathfrak{m}$-primary ideal. Assume $k$ is infinite. Let $N$ be a perfect $A$-module of dimension $t \geq 1$. Let $M$ be a MCM $A$-module. The function $n \rightarrow \ell(\text{Tor}^A_j(M, N/I^{n+1}N))$ is of p...
Let $F$ be a finitely generated free group, and let $L$ be a finitely presented $\mathbb{F}_q[F]$-module. We study the average number $\Lambda_L(n)$ of $\mathbb{F}_q[F]$-module homomorphisms from $L$ to an $\mathbb{F}_q[F]$-module of dimension $n$ over $\mathbb{F}_q$. We show that, for all sufficiently large $n$, the q...
J. de la Nuez González, Andrei Jaikin-Zapirain· 0 citations
Let $p$ and $\ell$ be distinct odd primes. For a finite extension $F/\mathbb{Q}_p$, the local Langlands correspondence states that there is a canonical bijection between irreducible, smooth representations of $\text{GL}_n(F)$ and $n$-dimensional, $\Phi$-semisimple Weil--Deligne representations of the Weil group $W_F$....
We show that von Neumann algebras of fundamental groups of closed orientable surfaces of genus $g\geq2$ are free group factors on $2g-1$generators. The key technical ingredient involves a proof that the element $w=ABA^{-1}B^{-1}$ of the free group $\mathbb{F}_{2}=\langle A,B\rangle$ is freely complemented in the group...
Let $\Omega$ be the Lorentz future cone in $\mathbb{R}^{d+1}$ with respect to the Lorentz product and let $T^M$ be the $M$-fold product of the future tube $T=\mathbb{R}^{d+1}+i\Omega$. The Lorentz group $\mathrm{SO}_0(1,d)$ acts diagonally on $T^M$, and its complexification $\mathrm{SO}(1,d)^\mathbb{C}$ acts on $\mathb...
Maxim Kukol· 0 citations
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