Research
My current mathematical interests like geometric group theory and low dimension topology. To be more precise, I study effective behavior of residual properties such as and conjugacy separability and their various generalizations to provide new asymptotic and algebraic invariants to groups of classic and contemporary interest. These include finite extensions of finitely generated nilpotent and solvable groups and groups of importance to low dimensional topology such as the mapping class group.
Collaborators: Alex Bishop, Meng-Che "Turbo" Ho, Ben McReynolds, Jonas Dere, Michal Ferov, Rob Kropholler, and Josiah Oh.
Collaborators: Alex Bishop, Meng-Che "Turbo" Ho, Ben McReynolds, Jonas Dere, Michal Ferov, Rob Kropholler, and Josiah Oh.
Solvable groups
Automorphism groups of solvable groups of finite abelian ranks
with Jonas Deré
preprint
The \(\mathbb{Q}\)-hull of a virtually polycyclic group \(\Gamma\) is an important tool to study its group of automophisms \(\text{Aut}(\Gamma)\) using linear algebraic groups. For example, it was used by Baues and Grünewald to prove that \(\text{Out}(\Gamma)\) is an arithmetic group, although \(\text{Aut}(\Gamma)\) is not for a generally virtually polycyclic group. The original definition due to Mostow was generalized to the class of virtually solvable groups of finite abelian ranks by Arapura and Nori to study Kähler manifolds, although other applications were not considered. This paper gives a new explicity construction of the \(\mathbb{Q}\)-algebraic hull for virtually solvable groups \(\Gamma\) of finite abelian ranks, taking into account the spectrum \(S\) of the group \(\Gamma\). As an application, we make a detailed study into the structure \(\text{Aut}(\Gamma)\) and show that several natural subgroups are \(S\)-arithmetic under the condition that \(\text{Fitt}(\Gamma)\) is \(S\)-arithmetic. Additionally, we demonstrate that the natural generalization of the aforementioned result by Baues and Grünewald fails even for most solvable Baumslag-Solitar groups. Among the other applications of our result is also a theorem showing that if \(\Gamma\) is strongly scale-invariant, it must be virtually nilpotent, giving a special case of a conjecture Nekrashevych and Pete and an extension of the previous known result for virtually polycyclic groups.
with Jonas Deré
preprint
The \(\mathbb{Q}\)-hull of a virtually polycyclic group \(\Gamma\) is an important tool to study its group of automophisms \(\text{Aut}(\Gamma)\) using linear algebraic groups. For example, it was used by Baues and Grünewald to prove that \(\text{Out}(\Gamma)\) is an arithmetic group, although \(\text{Aut}(\Gamma)\) is not for a generally virtually polycyclic group. The original definition due to Mostow was generalized to the class of virtually solvable groups of finite abelian ranks by Arapura and Nori to study Kähler manifolds, although other applications were not considered. This paper gives a new explicity construction of the \(\mathbb{Q}\)-algebraic hull for virtually solvable groups \(\Gamma\) of finite abelian ranks, taking into account the spectrum \(S\) of the group \(\Gamma\). As an application, we make a detailed study into the structure \(\text{Aut}(\Gamma)\) and show that several natural subgroups are \(S\)-arithmetic under the condition that \(\text{Fitt}(\Gamma)\) is \(S\)-arithmetic. Additionally, we demonstrate that the natural generalization of the aforementioned result by Baues and Grünewald fails even for most solvable Baumslag-Solitar groups. Among the other applications of our result is also a theorem showing that if \(\Gamma\) is strongly scale-invariant, it must be virtually nilpotent, giving a special case of a conjecture Nekrashevych and Pete and an extension of the previous known result for virtually polycyclic groups.
Effective Separability
\(\mathcal{C}\)-Hereditarily conjugacy separable groups and wreath products
with Alexander Bishop and Michal Ferov
preprint
We provide a necessary and sufficient condition for the restricted wreath product A≀B to be C-hereditarily conjugacy separable where \(\mathcal{C}\) is an extension-closed pseudo-vaiety of finite groups. Moreover, we prove that the Grigorchuk group is \(2\)-hereditarily conjugacy separable.
Survey on effective separability.
with Jonas Deré and Michal Ferov
to appear in "Geometric methods in group theory: papers dedicated to Ruth Charney" as part of the series Seminaires et Congres by the SMF
preprint
Separability for groups refers to the question which subsets of a group can be detected in its finite quotients. Classically, separability is studied in terms of which classes have a certain separability property, and this question is related to algorithmic problems in groups such as the word problem. A more recent perspective tries to study the order of the smallest finite quotient in which one detects the subset under consideration depending on its complexity, measured using the word norm on a finitely generated group. In this survey, we present what is currently known in the field of effective separability and give an overview of the open questions for several classes of groups.
Conjugacy depth function of wreath products of abelian groups
with Michal Ferov
J. Groups Complex. Cryptol. (1) 15 (2023), pp. 2:102:33
preprint
In this note, we complete the study of asymptotic behaviour of conjugacy separability of the general case of wreath products of finitely abelian groups where the base group is possibly infinite. In particular, we provide super-exponential upper and lower bounds for conjugacy separability of wreath products where the base group contains Z and, combining with previous work of the authors, we provide asymptotic bounds for conjugacy separability depth functions of all wreath products of finitely generated abelian groups. As an application, we give exponential lower bounds for infinitely many wreath products where the acting group is not necessarily abelian.
Quantifying conjugacy separability in wreath products of groups.
with Michal Ferov
Q. J. Math. 73 (2022), no. 4, 1555-1593
preprint
We study generalizations of conjugacy separability in restricted wreath products of groups. We provide an effective upper bound for C-conjugacy separability of a wreath product A≀B in terms of the C-conjugacy separability of A and B, the growth of C-cyclic subgroup separability of B, and the C-residual girth of B. As an application, we provide a characterisation of when A≀B is p-conjugacy separable. We use this characterisation to the provide for each prime p an example of wreath products with infinite base group that are p-conjugacy separable. We also provide asymptotic upper bounds for conjugacy separability for wreath products of nilpotent groups which include the lamplighter groups and provide asymptotic upper bounds for conjugacy separability of the free metabelian groups. Along the way, we provide a polynomial upper bound for the shortest conjugator between two elements of length at most n in a finitely generated nilpotent group.
Residual finiteness and strict distortion of cyclic subgroups of solvable groups
J. Algebra 546 (2020), 679-688
preprint
We provide polynomial lower bounds for residual finiteness of residually finite, finitely generated solvable groups that admit infinite order elements in the Fitting subgroup of strict distortion at least exponential. For this class of solvable groups which include polycyclic groups with a nontrivial exponential radical and the metabelian Baumslag-Solitar groups, we improve the lower bounds found in the literature. Additionally, for the class of residually finite, finitely generated solvable groups of infinite Prüfer rank that satisfy the conditions of our theorem, we provide the first nontrivial lower bounds.
Residual dimension of nilpotent groups
J. Group Theory 23 (2020), no. 5, 801-829
preprint
The functions \(F_G(n)\) measures the asymptotic behavior of residual finiteness for a finitely generated group \(G\). In previous work, the author claimed a characterization for \(F_N(n)\) when \(N\) is a finitely generated nilpotent group. However, a counterexample to the above claim was communicated to the author, and subsequently, the statement of the asymptotic characterization of \(F_N(n)\) is incorrect. In this article, we introduce new tools to provide lower asymptotic bounds for \(F_N(n)\) when \(N\) is a finitely generated nilpotent group. Moreover, we introduce a class of finitely generated nilpotent groups for which the upper bound of Pengitore_1 can be improved. Finally, we construct of a class of finitely generated nilpotent groups N for which the asymptotic behavior of \(F_N(n)\) can be fully characterized.
Effective Twisted Conjugacy Separability of Nilpotent Groups
with Jonas Deré
Math. Z. 292 (2019), no. 3-4, 763-790
preprint
This paper initiates the study of effective twisted conjugacy separability for finitely generated groups, which measures the complexity of separating distinct twisted conjugacy classes via finite quotients. The focus is on nilpotent groups, and our main result shows that there is a polynomial upper bound for twisted conjugacy separability. That allows us to study regular conjugacy separability in the case of virtually nilpotent groups, where we compute a polynomial upper bound as well. As another application, we improve the work of the second author by giving a possibly sharp upper bound for conjugacy separability for finitely generated nilpotent groups of nilpotency class \(2\).
Effective Separability of Finitely Generated Nilpotent Groups
New York J. Math. 24 (2018) 83-145
Available here
We give effective proofs of residual finiteness and conjugacy separability for finitely generated nilpotent groups. In particular, we give precise asymptotic bounds for a function introduced by Bou-Rabee that measures how large the quotients that are need to separate non-identity elements of bounded length from the identity which improves the work of Bou-Rabee. Similarly, we give polynomial upper and lower bounds for an analogous function introduced by Lawton, Louder, and McReynolds that measures how large the quotients that are need to separate pairs of distinct conjugacy classes of bounded work length using work Blackburn and Mal'tsev.
Effective Subgroup Separability of Nilpotent groups.
with Jonas Deré
J. Algebra 506 (2018), 489-508
preprint
This paper studies effective separability for subgroups of finitely generated nilpotent groups and more broadly effective subgroup separability of finitely generated nilpotent groups. We provide upper and lower bounds that are polynomial with respect to the logarithm of the word length for infinite index subgroups of nilpotent groups. In the case of normal subgroups, we provide an exact computation generalizing work of the second author. We introduce a function that quantifies subgroup separability, and we provide polynomial upper and lower bounds. We finish by demonstrating that our results extend to virtually nilpotent groups.
with Alexander Bishop and Michal Ferov
preprint
We provide a necessary and sufficient condition for the restricted wreath product A≀B to be C-hereditarily conjugacy separable where \(\mathcal{C}\) is an extension-closed pseudo-vaiety of finite groups. Moreover, we prove that the Grigorchuk group is \(2\)-hereditarily conjugacy separable.
Survey on effective separability.
with Jonas Deré and Michal Ferov
to appear in "Geometric methods in group theory: papers dedicated to Ruth Charney" as part of the series Seminaires et Congres by the SMF
preprint
Separability for groups refers to the question which subsets of a group can be detected in its finite quotients. Classically, separability is studied in terms of which classes have a certain separability property, and this question is related to algorithmic problems in groups such as the word problem. A more recent perspective tries to study the order of the smallest finite quotient in which one detects the subset under consideration depending on its complexity, measured using the word norm on a finitely generated group. In this survey, we present what is currently known in the field of effective separability and give an overview of the open questions for several classes of groups.
Conjugacy depth function of wreath products of abelian groups
with Michal Ferov
J. Groups Complex. Cryptol. (1) 15 (2023), pp. 2:102:33
preprint
In this note, we complete the study of asymptotic behaviour of conjugacy separability of the general case of wreath products of finitely abelian groups where the base group is possibly infinite. In particular, we provide super-exponential upper and lower bounds for conjugacy separability of wreath products where the base group contains Z and, combining with previous work of the authors, we provide asymptotic bounds for conjugacy separability depth functions of all wreath products of finitely generated abelian groups. As an application, we give exponential lower bounds for infinitely many wreath products where the acting group is not necessarily abelian.
Quantifying conjugacy separability in wreath products of groups.
with Michal Ferov
Q. J. Math. 73 (2022), no. 4, 1555-1593
preprint
We study generalizations of conjugacy separability in restricted wreath products of groups. We provide an effective upper bound for C-conjugacy separability of a wreath product A≀B in terms of the C-conjugacy separability of A and B, the growth of C-cyclic subgroup separability of B, and the C-residual girth of B. As an application, we provide a characterisation of when A≀B is p-conjugacy separable. We use this characterisation to the provide for each prime p an example of wreath products with infinite base group that are p-conjugacy separable. We also provide asymptotic upper bounds for conjugacy separability for wreath products of nilpotent groups which include the lamplighter groups and provide asymptotic upper bounds for conjugacy separability of the free metabelian groups. Along the way, we provide a polynomial upper bound for the shortest conjugator between two elements of length at most n in a finitely generated nilpotent group.
Residual finiteness and strict distortion of cyclic subgroups of solvable groups
J. Algebra 546 (2020), 679-688
preprint
We provide polynomial lower bounds for residual finiteness of residually finite, finitely generated solvable groups that admit infinite order elements in the Fitting subgroup of strict distortion at least exponential. For this class of solvable groups which include polycyclic groups with a nontrivial exponential radical and the metabelian Baumslag-Solitar groups, we improve the lower bounds found in the literature. Additionally, for the class of residually finite, finitely generated solvable groups of infinite Prüfer rank that satisfy the conditions of our theorem, we provide the first nontrivial lower bounds.
Residual dimension of nilpotent groups
J. Group Theory 23 (2020), no. 5, 801-829
preprint
The functions \(F_G(n)\) measures the asymptotic behavior of residual finiteness for a finitely generated group \(G\). In previous work, the author claimed a characterization for \(F_N(n)\) when \(N\) is a finitely generated nilpotent group. However, a counterexample to the above claim was communicated to the author, and subsequently, the statement of the asymptotic characterization of \(F_N(n)\) is incorrect. In this article, we introduce new tools to provide lower asymptotic bounds for \(F_N(n)\) when \(N\) is a finitely generated nilpotent group. Moreover, we introduce a class of finitely generated nilpotent groups for which the upper bound of Pengitore_1 can be improved. Finally, we construct of a class of finitely generated nilpotent groups N for which the asymptotic behavior of \(F_N(n)\) can be fully characterized.
Effective Twisted Conjugacy Separability of Nilpotent Groups
with Jonas Deré
Math. Z. 292 (2019), no. 3-4, 763-790
preprint
This paper initiates the study of effective twisted conjugacy separability for finitely generated groups, which measures the complexity of separating distinct twisted conjugacy classes via finite quotients. The focus is on nilpotent groups, and our main result shows that there is a polynomial upper bound for twisted conjugacy separability. That allows us to study regular conjugacy separability in the case of virtually nilpotent groups, where we compute a polynomial upper bound as well. As another application, we improve the work of the second author by giving a possibly sharp upper bound for conjugacy separability for finitely generated nilpotent groups of nilpotency class \(2\).
Effective Separability of Finitely Generated Nilpotent Groups
New York J. Math. 24 (2018) 83-145
Available here
We give effective proofs of residual finiteness and conjugacy separability for finitely generated nilpotent groups. In particular, we give precise asymptotic bounds for a function introduced by Bou-Rabee that measures how large the quotients that are need to separate non-identity elements of bounded length from the identity which improves the work of Bou-Rabee. Similarly, we give polynomial upper and lower bounds for an analogous function introduced by Lawton, Louder, and McReynolds that measures how large the quotients that are need to separate pairs of distinct conjugacy classes of bounded work length using work Blackburn and Mal'tsev.
- Mark Pengitore. Corrigendum to "Effective separability of finitely generated nilpotent groups"
New York J. Math. 24 (2018), 83--145
Available here
Effective Subgroup Separability of Nilpotent groups.
with Jonas Deré
J. Algebra 506 (2018), 489-508
preprint
This paper studies effective separability for subgroups of finitely generated nilpotent groups and more broadly effective subgroup separability of finitely generated nilpotent groups. We provide upper and lower bounds that are polynomial with respect to the logarithm of the word length for infinite index subgroups of nilpotent groups. In the case of normal subgroups, we provide an exact computation generalizing work of the second author. We introduce a function that quantifies subgroup separability, and we provide polynomial upper and lower bounds. We finish by demonstrating that our results extend to virtually nilpotent groups.
- Jonas Deré and Mark Pengitore. Corrigendum to "Effective Subgroup Separability of Nilpotent groups." [J. Algebra 506 (2018) 489-508]
J. Algebra 523 (2019), 365-367
Coarse Geometry
A coarse embedding theorem for homological filling functions.
with Rob Kropholler
Bull. Lond. Math. Soc. 54 (2022), no. 3, 876-890
preprint
We demonstrate under appropriate finiteness conditions that a coarse embedding induces an inequality of homological Dehn functions. Applications of the main results include a characterization of what finitely presentable groups may admit a coarse embedding into a hyperbolic group of geometric dimension 2, characterizations of finitely presentable subgroups of groups with quadratic Dehn function with geometric dimension 2, and to coarse embeddings of nilpotent groups into other nilpotent groups of the same growth and into hyperbolic groups.
Geometry of non-transitive graphs.
with Josiah Oh
Pacific J. Math. 317 (2022), no. 2, 423-440
preprint
In this note, we study non-transitive graphs and prove a number of results when they satisfy a coarse version of transitivity. Also, for each finitely generated group \(G\), we produce continuum many pairwise non-quasi-isometric regular graphs that have the same growth rate, number of ends, and asymptotic dimension as \(G\).
Coarse models of homogeneous spaces and translation-like actions.
with Ben McReynolds
Submitted
preprint
For finitely generated groups \(G\) and \(H\) equipped with word metrics, a translation-like action of \(H\) on \(G\) is a free action where each element of \(H\) moves elements of \(G\) a bounded distance. Translation-like actions provide a geometric generalization of subgroup containment. Extending work of Cohen, we show that cocompact lattices in a general semisimple Lie group \(\textbf{G}\) that is not isogenous to \(\text{SL}_2(\mathbb{R})\) admit translation-like actions by \(\mathbb{Z}^2\). This result follows from a more general result. Namely, we prove that any cocompact lattice in the unipotent radical \(\textbf{N}\) of the Borel subgroup \(\textbf{AN}\) of \(\textbf{G}\) acts translation-like on any cocompact lattice in \(\textbf{G}\). We also prove that for noncompact simple Lie groups \( \textbf{G} \), \( \textbf{H} \) with \(\textbf{H} \leq \textbf{G}\) and lattices \(\Gamma \leq \textbf{G}\) and \( \Delta \leq \textbf{H}\), that \(\Gamma/ \Delta\) is quasi-isometric to \(\textbf{G}/\textbf{H}\) where \(\Gamma/ \Delta \) is the quotient via a translation-like action of \(\Delta\) on \(\Gamma\).
Translation-like actions of nilpotent groups
with David Bruce Cohen
J. Topol. Anal. 11 (2019), no. 2, 405-426
preprint
We give a new obstruction to translation-like actions on nilpotent groups. Suppose we are given two finitely generated torsion free nilpotent groups with the same degree of polynomial growth, but non-isomorphic Carnot completions (asymptotic cones). We show that there exists no injective Lipschitz function from one group to the other. It follows that neither group can act translation-like on the other.
with Rob Kropholler
Bull. Lond. Math. Soc. 54 (2022), no. 3, 876-890
preprint
We demonstrate under appropriate finiteness conditions that a coarse embedding induces an inequality of homological Dehn functions. Applications of the main results include a characterization of what finitely presentable groups may admit a coarse embedding into a hyperbolic group of geometric dimension 2, characterizations of finitely presentable subgroups of groups with quadratic Dehn function with geometric dimension 2, and to coarse embeddings of nilpotent groups into other nilpotent groups of the same growth and into hyperbolic groups.
Geometry of non-transitive graphs.
with Josiah Oh
Pacific J. Math. 317 (2022), no. 2, 423-440
preprint
In this note, we study non-transitive graphs and prove a number of results when they satisfy a coarse version of transitivity. Also, for each finitely generated group \(G\), we produce continuum many pairwise non-quasi-isometric regular graphs that have the same growth rate, number of ends, and asymptotic dimension as \(G\).
Coarse models of homogeneous spaces and translation-like actions.
with Ben McReynolds
Submitted
preprint
For finitely generated groups \(G\) and \(H\) equipped with word metrics, a translation-like action of \(H\) on \(G\) is a free action where each element of \(H\) moves elements of \(G\) a bounded distance. Translation-like actions provide a geometric generalization of subgroup containment. Extending work of Cohen, we show that cocompact lattices in a general semisimple Lie group \(\textbf{G}\) that is not isogenous to \(\text{SL}_2(\mathbb{R})\) admit translation-like actions by \(\mathbb{Z}^2\). This result follows from a more general result. Namely, we prove that any cocompact lattice in the unipotent radical \(\textbf{N}\) of the Borel subgroup \(\textbf{AN}\) of \(\textbf{G}\) acts translation-like on any cocompact lattice in \(\textbf{G}\). We also prove that for noncompact simple Lie groups \( \textbf{G} \), \( \textbf{H} \) with \(\textbf{H} \leq \textbf{G}\) and lattices \(\Gamma \leq \textbf{G}\) and \( \Delta \leq \textbf{H}\), that \(\Gamma/ \Delta\) is quasi-isometric to \(\textbf{G}/\textbf{H}\) where \(\Gamma/ \Delta \) is the quotient via a translation-like action of \(\Delta\) on \(\Gamma\).
Translation-like actions of nilpotent groups
with David Bruce Cohen
J. Topol. Anal. 11 (2019), no. 2, 405-426
preprint
We give a new obstruction to translation-like actions on nilpotent groups. Suppose we are given two finitely generated torsion free nilpotent groups with the same degree of polynomial growth, but non-isomorphic Carnot completions (asymptotic cones). We show that there exists no injective Lipschitz function from one group to the other. It follows that neither group can act translation-like on the other.
Rational Growth
Rational growth for torus bundle groups of odd trace.
with Seongjun Choi and Meng-Che "Turbo" Ho
Proc. Edinburgh Math. Soc. (2) 65 (2022), no. 4, 1080-1132
preprint
A group is said to have a rational growth with respect to the generating set if the growth series is a rational polynomial. It was shown by Parry that a subset of torus bundle groups exhibits rational growth. We generalize this result to other torus bundle groups.
with Seongjun Choi and Meng-Che "Turbo" Ho
Proc. Edinburgh Math. Soc. (2) 65 (2022), no. 4, 1080-1132
preprint
A group is said to have a rational growth with respect to the generating set if the growth series is a rational polynomial. It was shown by Parry that a subset of torus bundle groups exhibits rational growth. We generalize this result to other torus bundle groups.
Cohomology
Rational cohomology and Zariski dense subgroups of solvable linear algebraic groups
with Milana Golich
preprint
In this article, we establish results concerning the cohomology of Zariski dense subgroups of solvable linear algebraic groups. We show that for an irreducible solvable Q-defined linear algebraic group \(\textbf{G}\), dthere exists an isomorphism between the cohomology rings with coefficients in a finite dimensional rational \(\textbf{G}\)-module \(M\) of the associated Q-defined Lie algebra \(\mathfrak{g}_{\mathbb{Q}}\) and Zariski dense subgroups \(\Gamma \leq \textbf{G}(\mathbb{Q})\) that satisfy the condition that they intersect the \(\mathbb{Q}\)-split maximal torus discretely. We further prove that the restriction map in rational cohomology from \(\textbf{G}\) to a Zariski dense subgroup \(\Gamma \leq \textbf{G}(\mathbb{Q})\) with coefficients in \(M\) is an injection. We then derive several results regarding finitely generated solvable groups of finite abelian rank and their representations on cohomology.
with Milana Golich
preprint
In this article, we establish results concerning the cohomology of Zariski dense subgroups of solvable linear algebraic groups. We show that for an irreducible solvable Q-defined linear algebraic group \(\textbf{G}\), dthere exists an isomorphism between the cohomology rings with coefficients in a finite dimensional rational \(\textbf{G}\)-module \(M\) of the associated Q-defined Lie algebra \(\mathfrak{g}_{\mathbb{Q}}\) and Zariski dense subgroups \(\Gamma \leq \textbf{G}(\mathbb{Q})\) that satisfy the condition that they intersect the \(\mathbb{Q}\)-split maximal torus discretely. We further prove that the restriction map in rational cohomology from \(\textbf{G}\) to a Zariski dense subgroup \(\Gamma \leq \textbf{G}(\mathbb{Q})\) with coefficients in \(M\) is an injection. We then derive several results regarding finitely generated solvable groups of finite abelian rank and their representations on cohomology.
Miscellaneous
-Charles R. Tolle, Mark Pengitore, Phase-Space Reconstruction: A Path Towards the Next Generation of Nonlinear Differential Equations Based Models and Its Implications Towards Non-Uniform Sampling Theory. IEEE ISRCS. 2009.
-Charles R. Tolle, Mark Pengitore, Phase-Space Reconstruction: A Path Towards the Next Generation of Nonlinear Differential Equations Based Models and Its Implications Towards Non-Uniform Sampling Theory. IEEE ISRCS. 2009.
