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To some within N.Y.U., Dr. Sexton is a hero who has transformed the university. The trustees have thanked him by elevating his salary to nearly $1.5 million from $773,000 and guaranteeing him retirement benefits of $800,000 a year.

But to others, he is an autocrat who treats all but a few anointed professors as hired help, ignoring their concerns, informing them of policies after the fact and otherwise running roughshod over American academic tradition, in which faculty members are partners in charting a university's course.

"He has a very evangelical sense of purpose," said Andrew Ross, a professor of social and cultural analysis, "that does not extend beyond the concept that the university should be an entity of his own making."

"I think," he added, "when other administrations see that they say, Well that's what leadership should be. And when faculty see that they say, That is not what university leadership should be. It's the style of a maverick C.E.O."

The debate over Dr. Sexton's presidency will come to a head this week. The faculty of the university's largest school, Arts and Science, has scheduled a five-day vote of no confidence. Given Dr. Sexton's international stature, the vote may serve as the most important referendum yet on the direction of American higher education.

The job description for university presidents has changed significantly in recent years. In a time of shrinking resources and rising costs, leaders must, of course, raise money; N.Y.U.'s recent $3 billion campaign set a national record. But they must also raise their institutions' profiles, forge strategic and business alliances, and plot digital strategy. Dr. Sexton, who declined to be interviewed for this article, has charged ahead on all those fronts, both in New York City and around the world.

The Global Network University that he established — with 12 international study centers in places like Accra, Ghana; Buenos Aires; and Paris, as well as a full degree-granting campus in Abu Dhabi and another one set to open next year in Shanghai — has turned N.Y.U. into a worldwide brand and broadened its students' horizons. At a State Department reception in 2011, Hillary Rodham Clinton, then secretary of state, praised his "vision to expand his university internationally while maintaining its reputation for excellence and academic freedom."

On a purely practical level, that global network has also allowed N.Y.U. to add students, including foreigners who pay full tuition, without having to find room for them in Greenwich Village. That is significant given N.Y.U.'s relatively small per-student endowment, which makes the university more dependent on tuition checks than other institutions in its academic league. (Students pay much less to attend N.Y.U. Abu Dhabi, thanks to subsidies from the local government, but that campus yields revenue for N.Y.U. in other ways, including an initial gift of $50 million from the government of Abu Dhabi.)

Back at home, Dr. Sexton won city approval for N.Y.U. 2031, a plan to build six million square feet of facilities in New York City, including almost two million on the two superblocks where 40 percent of the faculty members live, steps from Washington Square Park.

To a passionate, but as yet unquantified, segment of N.Y.U., each of those initiatives — and Dr. Sexton's willingness to go forward despite major faculty objections — shows he has placed his own ambition ahead of the best interests of the university.

Dr. Sexton expanded the faculty, bringing on celebrated thinkers and raising lackluster departments up to the top tier. But reflecting a nationwide trend, the majority of that growth has been in nontenure-track positions, including adjunct professors who earn just a few thousand dollars per course, and who might stay for one semester.

Adjunct professors do an especially large portion of the teaching at N.Y.U.'s 12 international centers, one of the reasons — along with the lack of academic freedom in Abu Dhabi and Shanghai — that critics describe the Global Network University as a cynical undertaking. Rebecca Karl, a history professor and a faculty senator, said she would support the idea of global education "if it weren't just about people paying an N.Y.U. New York tuition, then being shipped off to Prague where you can house and educate the kids for a fraction of the cost that it takes in New York and then N.Y.U. pockets the difference."

J. Anthony Movshon, the director of the Center for Neural Science, said he supported of Dr. Sexton and was generally optimistic about the global push. But he said it "has created burdens for departments and individual faculty who are asked to cover the needs of those campuses as they grow."

"It's been complicated," he added, "and disruptive."

Without question the most controversial of Dr. Sexton's policies has been N.Y.U. 2031, the plan for expansion in New York City. Though everyone agrees that the university is squeezed for space, in a survey last year, 716 faculty members, out of 1,126 responding, disapproved of the plan. The Stern Business School and the economics department jointly expressed concerns that its cost would crush the university, forcing tuition increases and lower teacher-student ratios.

Faculty opponents have joined neighborhood groups in a lawsuit to block the plan; last week, Henry J. Stern, the former New York City parks commissioner, signed on, saying the plan unlawfully appropriated public park space.

Those who favor N.Y.U. 2031 tend to describe the opposition as a wishful — and belated — attempt to move the construction to someone else's backyard. But even some supporters concede that the administration's approach made the conflict worse. In testimony in April before the City Planning Commission, Dalton Conley, then the dean of social sciences, strongly endorsed the plan, but added, "Had more faculty been involved in the process itself and had it been less driven by administration officials, you would see few if any professors testifying against it here today."

In person, Dr. Sexton comes across as folksy. A former debating champion, he has mastered the gentler rhetorical skill of conveying his thoughts through personal anecdotes. Like the hugs with which he greets even strangers, these stories can disarm opponents, either by setting a jovial tone for a conversation by deftly guiding it away from any minefields or simply by running out the clock. But with a vote of no confidence on the schedule, the conversation, at least for the moment, seems to have gotten away from him.

But according to B. Robert Kreiser of the American Association of University Professors, no-confidence votes are often unsuccessful, and can even backfire "because boards rally around the president, often extending their contracts for years."

If that happens at N.Y.U., said Andrew Delbanco, a professor at Columbia University and the author of "College: What It Was, Is, and Should Be," it "will embolden other presidents to disregard, or at least discount, the faculty as a merely retrograde force."

Regardless, Dr. Sexton has begun adjusting his approach to the faculty — as a means of learning from experience, say supporters like Martin Lipton, chairman of N.Y.U.'s board of trustees, or of buying off dissent, his critics say. The e-mail Dr. Sexton sent his faculty shortly before Christmas struck an uncharacteristically somber and humble tone.

"We have taken some steps to provide for improved faculty input and critique," he wrote. "I know more must be done, and during the winter recess I will be reflecting on how I can help to achieve that."

For Professor Karl, however, no such adjustments can alter the fundamental questions. "Who and what is the university for?" she asks. "And in which ways should that get shaped? We all recognize that we're in a brave new world."