Benjamin Martin
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Benjamin Martin has a total of 26 co-authors. Here are the top co-authors by number of shared publications:
- Michael Bate
Department of Mathematics, University of York, York YO10 5DD, United Kingdom
18 shared publications - Gerhard Roehrle
Ruhr-Universität Bochum
14 shared publications - Sören Böhm
Fakultät für Mathematik, Ruhr-Universität Bochum, Bochum, Germany
4 shared publications Gerhard Röhrle
3 shared publications- Sebastian Herpel
Ruhr-Universität Bochum
3 shared publications - Alastair Litterick
School of Mathematics, Statistics and Actuarial Science, University of Essex, Colchester, United Kingdom
2 shared publications
Latest Publications
- Open Access
- 18 June 2025
- Published by Springer Nature in Geometriae Dedicata
- Vol. 219 (4) , 1-25
Essential tori in 3–manifolds not detected in any characteristic
Infinite families of 3–dimensional closed graph manifolds and closed Seifert fibered spaces are exhibited, each member of which contains an essential torus not detected by ideal points of the variety of $${{\,\textrm{SL}\,}}_2(\mathbb {F})$$ –characters over any algebraically closed field $$\mathbb {F}$$ .- 18 June 2025
- Published by Springer Nature in Geometriae Dedicata
- Vol. 219 (4) , 1-25
Open AccessEssential tori in 3–manifolds not detected in any characteristic
Infinite families of 3–dimensional closed graph manifolds and closed Seifert fibered spaces are exhibited, each member of which contains an essential torus not detected by ideal points of the variety of $${{\,\textrm{SL}\,}}_2(\mathbb {F})$$ –characters over any algebraically closed field $$\mathbb {F}$$ . - Open Access
- 2 June 2025
- Published by Mathematical Sciences Publishers in Pacific Journal of Mathematics
- Vol. 337 (1) , 1-24
G-complete reducibility and saturation
Let H ⊆ G be connected reductive linear algebraic groups defined over an algebraically closed field of characteristic p > 0. In our first main theorem we show that if a closed subgroup K of H is H-completely reducible, then it is also G-completely reducible in the sense of Serre, under some restrictions on p, generalising the known case for G = GL(V). Our proof uses R. W. Richardson's notion of reductive pairs to reduce to the GL(V) case. We study Serre's notion of saturation and prove that saturation behaves well with respect to products and regular subgroups. Our second main theorem shows that if K is H-completely reducible, then the saturation of K in G is completely reducible in the saturation of H in G (which is again a connected reductive subgroup of G), under suitable restrictions on p, again generalising the known instance for G = GL(V). We also study saturation of finite subgroups of Lie type in G. We show that saturation is compatible with standard Frobenius endomorphisms, and we use this to generalise a result due to Nori from 1987 in the case G = GL(V).
- 2 June 2025
- Published by Mathematical Sciences Publishers in Pacific Journal of Mathematics
- Vol. 337 (1) , 1-24
Open AccessG-complete reducibility and saturation
Let H ⊆ G be connected reductive linear algebraic groups defined over an algebraically closed field of characteristic p > 0. In our first main theorem we show that if a closed subgroup K of H is H-completely reducible, then it is also G-completely reducible in the sense of Serre, under some restrictions on p, generalising the known case for G = GL(V). Our proof uses R. W. Richardson's notion of reductive pairs to reduce to the GL(V) case. We study Serre's notion of saturation and prove that saturation behaves well with respect to products and regular subgroups. Our second main theorem shows that if K is H-completely reducible, then the saturation of K in G is completely reducible in the saturation of H in G (which is again a connected reductive subgroup of G), under suitable restrictions on p, again generalising the known instance for G = GL(V). We also study saturation of finite subgroups of Lie type in G. We show that saturation is compatible with standard Frobenius endomorphisms, and we use this to generalise a result due to Nori from 1987 in the case G = GL(V).
- Open Access
- 26 May 2025
- Published by Mathematical Sciences Publishers in Pacific Journal of Mathematics
- Vol. 336 (1-2) , 29-61
On good A1 subgroups, Springer maps, and overgroups of distinguished unipotent elements in reductive groups
- 26 May 2025
- Published by Mathematical Sciences Publishers in Pacific Journal of Mathematics
- Vol. 336 (1-2) , 29-61
Open AccessOn good A1 subgroups, Springer maps, and overgroups of distinguished unipotent elements in reductive groups
- Open Access
Essential tori in 3-manifolds not detected in any characteristic
Infinite families of 3-dimensional closed graph manifolds and closed Seifert fibered spaces are exhibited, each member of which contains an essential torus not detected by ideal points of the variety of $\text{SL}_2(\mathbb{F})$-characters over any algebraically closed field $\mathbb{F}$.Open AccessEssential tori in 3-manifolds not detected in any characteristic
Infinite families of 3-dimensional closed graph manifolds and closed Seifert fibered spaces are exhibited, each member of which contains an essential torus not detected by ideal points of the variety of $\text{SL}_2(\mathbb{F})$-characters over any algebraically closed field $\mathbb{F}$. - Open Access
- 14 May 2024
- Published by Cambridge University Press (CUP) in Proceedings of the Edinburgh Mathematical Society
- Vol. 67 (3) , 830-837
Powers of commutators in linear algebraic groups
Let ${\mathcal G}$ be a linear algebraic group over k, where k is an algebraically closed field, a pseudo-finite field or the valuation ring of a non-archimedean local field. Let $G= {\mathcal G}(k)$ . We prove that if $\gamma\in G$ such that γ is a commutator and $\delta\in G$ such that $\langle \delta\rangle= \langle \gamma\rangle$ then δ is a commutator. This generalises a result of Honda for finite groups. Our proof uses the Lefschetz principle from first-order model theory.- 14 May 2024
- Published by Cambridge University Press (CUP) in Proceedings of the Edinburgh Mathematical Society
- Vol. 67 (3) , 830-837
Open AccessPowers of commutators in linear algebraic groups
Let ${\mathcal G}$ be a linear algebraic group over k, where k is an algebraically closed field, a pseudo-finite field or the valuation ring of a non-archimedean local field. Let $G= {\mathcal G}(k)$ . We prove that if $\gamma\in G$ such that γ is a commutator and $\delta\in G$ such that $\langle \delta\rangle= \langle \gamma\rangle$ then δ is a commutator. This generalises a result of Honda for finite groups. Our proof uses the Lefschetz principle from first-order model theory. - Open Access
Powers of commutators in linear algebraic groups
Let ${\mathscr G}$ be a linear algebraic group over $k$, where $k$ is an algebraically closed field, a pseudo-finite field or the valuation ring of a nonarchimedean local field. Let $G= {\mathscr G}(k)$. We prove that if $\gamma, \delta\in G$ such that $\gamma$ is a commutator and $\langle \delta\rangle= \langle \gamma\rangle$ then $\delta$ is a commutator. This generalises a result of Honda for finite groups. Our proof uses the Lefschetz Principle from first-order model theory.Open AccessPowers of commutators in linear algebraic groups
Let ${\mathscr G}$ be a linear algebraic group over $k$, where $k$ is an algebraically closed field, a pseudo-finite field or the valuation ring of a nonarchimedean local field. Let $G= {\mathscr G}(k)$. We prove that if $\gamma, \delta\in G$ such that $\gamma$ is a commutator and $\langle \delta\rangle= \langle \gamma\rangle$ then $\delta$ is a commutator. This generalises a result of Honda for finite groups. Our proof uses the Lefschetz Principle from first-order model theory. - Open Access
- 1 December 2023
- Published by Springer Nature in European Journal of Mathematics
- Vol. 9 (4) , 1-27
Complete reducibility for Lie subalgebras and semisimplification
Let G be a connected reductive linear algebraic group over a field k. Using ideas from geometric invariant theory, we study the notion of G-complete reducibility over k for a Lie subalgebra $${\mathfrak {h}}$$ of the Lie algebra $${\mathfrak {g}} = \hbox {Lie} (G)$$ of G and prove some results when $${\mathfrak {h}}$$ is solvable or $$\textrm{char}\hspace{0.55542pt}(k)= 0$$ . We introduce the concept of a k-semisimplification $${\mathfrak {h}}'$$ of $${\mathfrak {h}}$$ ; $${\mathfrak {h}}'$$ is a Lie subalgebra of $${\mathfrak {g}}$$ associated to $${\mathfrak {h}}$$ which is G-completely reducible over k. This is the Lie algebra counterpart of the analogous notion for subgroups studied earlier by the first, third and fourth authors. As in the subgroup case, we show that $${\mathfrak {h}}'$$ is unique up to $$\textrm{Ad}\hspace{0.55542pt}(G(k))$$ -conjugacy in $${\mathfrak {g}}$$ . Moreover, we prove that the two concepts are compatible: for H a closed subgroup of G and $$H'$$ a k-semisimplification of H, the Lie algebra $$\textrm{Lie}\hspace{0.55542pt}(H')$$ is a k-semisimplification of $$\textrm{Lie}\hspace{0.55542pt}(H)$$ .- 1 December 2023
- Published by Springer Nature in European Journal of Mathematics
- Vol. 9 (4) , 1-27
Open AccessComplete reducibility for Lie subalgebras and semisimplification
Let G be a connected reductive linear algebraic group over a field k. Using ideas from geometric invariant theory, we study the notion of G-complete reducibility over k for a Lie subalgebra $${\mathfrak {h}}$$ of the Lie algebra $${\mathfrak {g}} = \hbox {Lie} (G)$$ of G and prove some results when $${\mathfrak {h}}$$ is solvable or $$\textrm{char}\hspace{0.55542pt}(k)= 0$$ . We introduce the concept of a k-semisimplification $${\mathfrak {h}}'$$ of $${\mathfrak {h}}$$ ; $${\mathfrak {h}}'$$ is a Lie subalgebra of $${\mathfrak {g}}$$ associated to $${\mathfrak {h}}$$ which is G-completely reducible over k. This is the Lie algebra counterpart of the analogous notion for subgroups studied earlier by the first, third and fourth authors. As in the subgroup case, we show that $${\mathfrak {h}}'$$ is unique up to $$\textrm{Ad}\hspace{0.55542pt}(G(k))$$ -conjugacy in $${\mathfrak {g}}$$ . Moreover, we prove that the two concepts are compatible: for H a closed subgroup of G and $$H'$$ a k-semisimplification of H, the Lie algebra $$\textrm{Lie}\hspace{0.55542pt}(H')$$ is a k-semisimplification of $$\textrm{Lie}\hspace{0.55542pt}(H)$$ . - Open Access
Complete reducibility for Lie subalgebras and semisimplification
Let $G$ be a connected reductive linear algebraic group over a field $k$. Using ideas from geometric invariant theory, we study the notion of $G$-complete reducibility over $k$ for a Lie subalgebra $\mathfrak h$ of the Lie algebra $\mathfrak g = Lie(G)$ of $G$ and prove some results when $\mathfrak h$ is solvable or $char(k)= 0$. We introduce the concept of a $k$-semisimplification $\mathfrak h'$ of $\mathfrak h$; $\mathfrak h'$ is a Lie subalgebra of $\mathfrak g$ associated to $\mathfrak h$ which is $G$-completely reducible over $k$. This is the Lie algebra counterpart of the analogous notion for subgroups studied earlier by the first, third and fourth authors. As in the subgroup case, we show that $\mathfrak h'$ is unique up to $Ad(G(k))$-conjugacy in $\mathfrak g$. Moreover, we prove that the two concepts are compatible: for $H$ a closed subgroup of $G$ and $H'$ a $k$-semisimplification of $H$, the Lie algebra $Lie(H')$ is a $k$-semisimplification of $Lie(H)$.Open AccessComplete reducibility for Lie subalgebras and semisimplification
Let $G$ be a connected reductive linear algebraic group over a field $k$. Using ideas from geometric invariant theory, we study the notion of $G$-complete reducibility over $k$ for a Lie subalgebra $\mathfrak h$ of the Lie algebra $\mathfrak g = Lie(G)$ of $G$ and prove some results when $\mathfrak h$ is solvable or $char(k)= 0$. We introduce the concept of a $k$-semisimplification $\mathfrak h'$ of $\mathfrak h$; $\mathfrak h'$ is a Lie subalgebra of $\mathfrak g$ associated to $\mathfrak h$ which is $G$-completely reducible over $k$. This is the Lie algebra counterpart of the analogous notion for subgroups studied earlier by the first, third and fourth authors. As in the subgroup case, we show that $\mathfrak h'$ is unique up to $Ad(G(k))$-conjugacy in $\mathfrak g$. Moreover, we prove that the two concepts are compatible: for $H$ a closed subgroup of $G$ and $H'$ a $k$-semisimplification of $H$, the Lie algebra $Lie(H')$ is a $k$-semisimplification of $Lie(H)$. - Open Access
Smoothness of stabilisers in generic characteristic
Let $R$ be a commutative unital ring. Given a finitely presented affine $R$-group scheme $G$ acting on a separated scheme $X$ of finite type over $R$, we show that there is a prime $p_0$ such that for any $R$-algebra $k$ which is an algebraically closed field of characteristic $p\geq p_0$, the centraliser in $G_k$ of any closed subscheme of $X_k$ is smooth. When $X$ is not necessarily separated we show similarly that for any closed subscheme $Y \subseteq X$ there is a $p_1$ depending on $Y$ such that when $k$ has characteristic $p \geq p_1$ the normaliser of $Y$ in $G_k$ is smooth. We prove these results using the Lefschetz principle together with careful application of Gr\"obner basis techniques, and using a suitable notion of the complexity of an action. We apply our results to demonstrate that the Kostant-Kirillov-Souriau theorem holds for Lie algebras of algebraic groups in large positive characteristics. In particular, every such Lie algebra decomposes as a disjoint union of symplectic varieties, each of which is a coadjoint orbit.Open AccessSmoothness of stabilisers in generic characteristic
Let $R$ be a commutative unital ring. Given a finitely presented affine $R$-group scheme $G$ acting on a separated scheme $X$ of finite type over $R$, we show that there is a prime $p_0$ such that for any $R$-algebra $k$ which is an algebraically closed field of characteristic $p\geq p_0$, the centraliser in $G_k$ of any closed subscheme of $X_k$ is smooth. When $X$ is not necessarily separated we show similarly that for any closed subscheme $Y \subseteq X$ there is a $p_1$ depending on $Y$ such that when $k$ has characteristic $p \geq p_1$ the normaliser of $Y$ in $G_k$ is smooth. We prove these results using the Lefschetz principle together with careful application of Gr\"obner basis techniques, and using a suitable notion of the complexity of an action. We apply our results to demonstrate that the Kostant-Kirillov-Souriau theorem holds for Lie algebras of algebraic groups in large positive characteristics. In particular, every such Lie algebra decomposes as a disjoint union of symplectic varieties, each of which is a coadjoint orbit. - Open Access
- 24 February 2022
- Published by Cambridge University Press (CUP) in Forum of Mathematics, Sigma
- Vol. 10
Overgroups of regular unipotent elements in reductive groups
We study reductive subgroups H of a reductive linear algebraic group G – possibly nonconnected – such that H contains a regular unipotent element of G. We show that under suitable hypotheses, such subgroups are G-irreducible in the sense of Serre. This generalises results of Malle, Testerman and Zalesski. We obtain analogous results for Lie algebras and for finite groups of Lie type. Our proofs are short, conceptual and uniform.
- 24 February 2022
- Published by Cambridge University Press (CUP) in Forum of Mathematics, Sigma
- Vol. 10
Open AccessOvergroups of regular unipotent elements in reductive groups
We study reductive subgroups H of a reductive linear algebraic group G – possibly nonconnected – such that H contains a regular unipotent element of G. We show that under suitable hypotheses, such subgroups are G-irreducible in the sense of Serre. This generalises results of Malle, Testerman and Zalesski. We obtain analogous results for Lie algebras and for finite groups of Lie type. Our proofs are short, conceptual and uniform.